I think the best way to keep this character feeling relatable and not like a two-faced jerk is to have them acknowledge that their thoughts are bad, but they're trying their best to fix them.
For example, a man accidentally stomps on their foot.
Their initial thought is something harsh like Ow, why isn't this idiot watching where he's going?
Then the character catches themselves and thinks No, it's not that big a deal. I wasn't paying attention either. Take a couple of nice deep breaths and handle the situation calmly.
If the person who stepped on his foot apologizes then he can say "It's okay. I'm sure it was an accident." And if the other person just keeps walking by he'll just decide not to make a scene.
I'm sure we all have times where we get annoyed by something but have to censor ourselves because we know what we're really thinking could hurt someone's feelings. Just emphasize the second part.
Basically, this is a character who has a short temper but is trying his best to be patient with people anyway. Honestly, I think that's pretty admirable. They don't want to hurt anyone's feelings no matter how bad a day they personally are having.
You could also make this character more rootable by giving them an absolutely horrible day and showing they still make an effort to be nice to people regardless. They lost their job, broke up with their significant other, and broke their leg all on the same day, but no matter how hopeless and angry they feel they still outwardly smile. That shows a lot of moral character on their part.
You could also surround them with horrible or just plain annoying people. Show this character has the patience of a saint by surrounding them with some of the most obnoxious people imaginable. Their parents, friends, and co-workers are all totally insufferable to be around, but they have to put up with it anyway. Every day is a fight to keep their sanity, yet they somehow manage to keep a positive attitude in spite of this.
If the audience believes this character has good reason to be mad, then they'll be even more impressed that they manage to keep their cool regardless.