I need a formulation that clearly indicates that a binding choice is made by pressing a button. The expression is written on a Button, so it has to be concise.
I currently use "Select X bindingly", which sounds awful.
Any ideas are appreciated.
You could opt for "Permanently select X"; permanent sounds more natural than binding in this case.
However (but this is more of a UX answer than an English answer), consider other alternatives to make sure the users realize they make a binding choice, such as a longer sentence just above the button (maybe explaining why the choice is binding), or a similarly worded confirmation popup.
How about a button saying "Commit!" (I agree with Glorfindel, though, that there should be some kind of accompanying clarification.)
I would suggest something that will catch the reader's attention and trigger their legal reflexes, so that they understand that it is serious. For example, "Sign the contract" would make anyone pause and think before clicking.
If it isn't a contract, but a formal promise to do something, you could use the verb "Pledge" (especially if their lives, fortunes and sacred honor are on the line).