You normally wouldn't start by writing the complete story from start to finish and then proceed to start developing the other parts of a game: gameplay, art, music, ...
Regarding the story the normal process seems to be that you would start with a high-level concept - this article called Do you want to write video games? calls that part of the job the narrative designer. The job is to get a feeling for the rough sketches - which characters at which point go on what kind of quest? Later the writer comes and gets the design for a quest - it's up to him what each character says and how exactly they interact with each other. See also this answer to my question How do big creative writing projects with multiple people work, preferably in the videogame industry? It will give you more insight into the general process of bigger writing projects.
That means you wouldn't have beta readers and you wouldn't finish the story and then start with the rest. You would have a high-level concept, defining the most important characters and the steps they take towards their goal.
This high-level document is what you could give your beta-readers - give them your rough sketch as what it is: a rough sketch. If you want to give them a gaming experience you could for example use the program you mentioned, Twine, which is a tool for creating interactive HTML-based gamebooks (Choose-your-own-adventure-book-style games). Another option would be Choicescript, which S. Mitchell showed me in my question Where can I find resources about writing “Choose your own adventure”-style books? Or you could script a text-based console game if you know the basics of a programming language.
But it would be more than sufficient enough to simply leave it as a text document and show it to your beta readers. They will see that it's still a sketch and then they will give you high-level feedback so that you don't get caught up in small details like typos and grammar, when you didn't even start with your game.
While working on the game all parts will evolve towards the final product and you can start to give people finer versions of your script, for example by providing them with simple sample missions that you have already fleshed out a bit more. If you want to get feedback on the story first before starting to work on the gaming experience you can simply create the detailed writing part that you will use as a draft for the final experience and give that to your audience before you start working on it. While your beta-readers are giving you feedback on which parts still need work you can flesh out the next parts.
That's no different from getting feedback for a novel or other kind of... anything honestly. You don't want to be completely done, having spent ages on small scale details, only to have someone point out a seeminlgy obvious flaw that make you wish you could view your own creation from the eyes of someone else so that you would have seen this catastrophe before that means you have just wasted precious days, weeks or even months of your precious time - when in fact the very same person might have given you the very same feedback at the start with only the high-level document.
Start with an outline and give that to your beta-readers.
Work on the first fleshed out stuff while you are collecting and categorizing feedback.
Start fleshing out a few chapters and give each to your beta-readers. This step includes first steps in other parts of the game than writing - it encompasses writing, gaming, music, ...
Work on the next fleshed out chapters while you are collecting feedback for the first ones.
After each of the previous steps you should make a round where you re-evaluate what you have done and where you are standing.
Here is a nice quote from the article I linked that applies to so many things in life and this is one of those things: "The reason this is 80 percent of your job is because you will spend the majority of your time fixing shit."