There are a few things that need to be made clear:
1: How are females seen in the story's society?
2: What age is one considered adult and can, therefore, legally decide for themselves?
Let's go over some possible scenarios, shall we?
1. Women are inferior to men (despite some usefulness) and are never legally able to choose for themselves
2. Women are inferior to men (despite some usefulness) but are legally adults at about 18-21
3. Women are inferior to men (despite some usefulness), nevertheless they are legally adults at about 15
4. Women are not supposedly inferior to men (1) (similar to nowadays) and are legally adults at about 18-21
5. Women are not supposedly inferior to men (1) (similar to nowadays) and are legally adults at about 15
These scenarios basically boil down to two possibilities:
a) the father doesn't have the legal right to stop her from going to court.
If he doesn't, he can worry and complain all he wants - it won't stop her daughter from fulfilling her dream and you can play his worry as a reason for plotting (to make sure she is safe).
b) the father has the legal right to stop her from going to court.
If he does have the legal right to stop her, though, one must focus on his legitimate reasons.
Since we want legitimate reasons, let's ignore for now a need to over-protect.
What exactly are the cons, here?
You say that he has trouble seeing his daughter as 'grown up'. Honestly, so do I. I work with children and teenagers (middle class). The most mature fifteen and sixteen years old are still naïve, very positive and idealistic. Even those who focus on the problems around them and wish to do something, they are still too inexperienced to grasp the deeper nuances of the situations.
It is interesting to note that, in the 14th century, a female was deemed physically grown up at 12 (or whenever mentruation showed up) while a male was deemed physically grown up at 14 (when a prince could tell his regent to step down and then take the full responsibilities or his position). Nevertheless, it was still believed that a man would not be fully mentally mature before he was 20-25. There was a debate whether women could reach full mental maturity; few believed it likely.
The bottom line is: at 15, the girl may look grown up (ie, physically she'd be seen more as a woman and less as a girl), but mentally and psychologically she would still be inexperienced and, therefore, naïve.
Ah, but she is wise beyond her years and extremely intelligent!
Unfortunately, being secluded away from politics and intrigues, she is likely to be unprepared for dealing with expert manipulators. If someone decides to use her for their ends, she will be an easier target than a young woman who grew up witnessing such plots.
Ah, but she is well read and aware of the range of evil people as well as what they are capable of!
Knowing the theory and spotting the manipulators in real life, under pressure and overwhelmed by the different lifestyles... even psychologists fall for plays of particularly cunning teenagers! And we're talking about adults who specialised in understanding and dealing with all sorts of personalities.
You point out the court is truly dangerous and that her father has enemies who would wish to use her against him. The father would have a legitimate reason to fear for his daughter well-being, be it sexually, emotionally, socially or even in terms of life-and-death.
The Daughter's Perspective
This is the easy one. She has suffered physically and emotionally due to her disability. She is sequestered in a far off estate, probably lonely, and hungers for something that stimulates her intelligence. She longs for a challenge that will let her prove her worth, both to others (maybe particularly to her distant, protective father) and to herself. She is not a naïve little girl! She's well aware of the dangers but she is also very much aware she can handle them... carefully, of course. Maybe with some assistance (perhaps even from her beloved father!). But she is confident she can help herself, her father, and, who knows!, the entire kingdom.
The Father's Perspective
Now put yourself in the father's shoes. You failed your daughter when she was but a little child. She nearly died and became a cripple! Because you failed her. What worthy man will look at her as a full woman? What will all those snakes and vultures think as they set their sights on her? She is intelligent - you know it well and with pride! - but are you really going to let her come into a viper's nest? Will you stand and look as your enemies gleefully plot to destroy her and, to make it worse, because of who you are? Will you fail her again? Seriously?
No! You will stop this nonsense! Of course she is happy with the honour! Of course she longs for something more than her secluded lifestyle! She is an intelligent young woman. You know she wishes to meet like-minded people and make herself useful. Well, you'll arrange a worthy position for her. You will talk to an ally of yours and you will have him accept your daughter as a lady-in-waiting to his wife. Your friend is known for his library and his intellectual prowess. Surely he will value your daughter's intellectual ability!
Ah, but perhaps it's out of your hands! The powers that be liked the suggestion and they will have your daughter come in. What can you do to protect her?
First, you must face the one who thought of giving your daughter such a position. You will remind him of what happened to Lord X and to Lady Y, who were so shamelessly tricked into trouble. One was exiled, the other was sentenced to death. Now, it is time to make him understand: should anything happen to your daughter, he'll be begging for a swift death. And it is not a vain threat!
Secondly, you will start studying those who will be working close to your daughter, within her functions. You'll plot out who is an enemy, who is likely to become a pawn and who you can use to watch over her. You must also deploy an able agent who, without anyone's suspicion, can be constantly by her side, protecting her in every way possible.
And, the most difficult thing of all, you must hide how much you fear and suffer for her! Your enemies will use it against you... and will do so through that innocent child of yours.
Less dramatically, now.
Make the father's fears be completely justified. Make sure there are examples of people who have been disgraced through intrigue, and make that disgrace be fatal - whether literally or socially. If there is a 100% chance the girl will be targeted by the most cunning minds so as to have her or her father or both exiled (or executed) while his enemies worm their way into the king's brains, then allowing her to go to court is almost reckless and negligent. Would you allow your child to walk into something with those odds?
On the other hand, make sure he is not only aware of her skills and abilities but also proud of them, even if he fears those skills are not enough. As my father once said when I called him out on his apparent distrust of my decision-making abilities: 'I have absolute trust in you. It's everyone else I don't trust."
Make sure he understands her desire to 'go into the world and make her mark'. He can even long to see her do just that so those who belittle her and patronise her can finally see her for the strong-minded person she is. Just not in the middle of vipers and vultures who would crush her with a smile!
These two points will make the readers not only sympathise with the father but also feel he would be a negligent father if he simply okayed her going without a fight.
So basically, he's being a mother hen.
Make sure no one sees him as a mother hen. Mother hens over-protect: they see an unlikely danger as impending doom and would rather their child doesn't learn to ride a bike for fear of a scraped knee. You can sympathise with a mother-hen for the first few minutes (or seconds), then you want her to open her eyes and get a reality-check.
Instead, he keeps sliding into "annoying, patronizing, ableist misogynist" territory.
Do not let him wander over many topics. There are only two things that matter:
1: his daughter is a wonderful young woman and will become a great woman in time (no misogyny here)
2: there are very real and very fatal dangers - which have befallen others more experienced - waiting for her in the court (if it's real, his fear cannot be annoying or patronising)
However, make sure he doesn't whine about it: it will definitely be annoying. Show his determination to keep her from harm while never losing faith in her abilities.
In a particular scene, I'm having him confront the person who offered the girl the position. What he's saying should be the words of a worried father. Instead, he comes off as a very unsympathetic person.
I can only suggest you keep trying until it comes off right. I'll offer a mid-argument suggestion:
"She is a very intelligent young woman, John. You should have more faith in her!
"You think I don't know that? But she's still just a child! Look at the girls her age in the court! The way they flirt and prize nothing but appearance. They will take any chance they can to humilliate and belittle her. And those lover-boys! What experience does my Matilda have with such filthy men? She is intelligent, yes! But she is still a young woman and don't you tell me young women's hearts don't end up succumbing to at least one of those blasted Don Juans!
And if it were only that, George! Matilda won't care for those chicken headed girls, no matter how much they try to humilliate her. She won't care for those empty-headed cokerels, no matter how hard they try to ensnare her! But what happens if Lady Catherine sets her eyes on her? You know the treachery of that poisonous tongue. She could make the Devil believe she's more innocent than a newborn. Look how she disgraced Lady Eveline, the poor woman. She was ten years older than my Matilda and she grew up in the city. She was not one to be fooled and was intelligent in her own right. What good did that do to her?"
Notice that in the father's first paragraph he focuses on the typical dangers for a young woman - attacks from her peers and from the beaus - but then he himself belittles those dangers by claiming his daughter is above them. The paradox of saying A will happen followed by A will never happen can be played for great effect to show his conflicting emotions: the fear of the dangers and his faith in her.
Then comes the real danger. There was a previous victim who is also pointed out as intelligent (no misogyny here), but neither intelligence (which his daughter has) nor experience (which his daughter doesn't have) saved her. It underlines how dangerous the court is and it justify his fears, making the reader also fear for the safety of the girl.
(1) while scientifically women are recognised as equal to men, a lot of people in the western world still have trouble realising it in real life