Okay. So, I looked at the code. The actual issue is that the in-game tooltip describing why the button is unclickable is extremely misleading. It heavily implies that you must not be wearing any clothing at all, which is very much not the case -- you must be wearing clothing, but not any removable bindings. It doesn't actually care about unremovable bindings. So I could not figure out why the fuck it wasn't working, because I was trying to do what the tooltip said to do and not what the actual code of the game asks for. So yeah... that's an ultra misleading tooltip and should definitely be rewritten.
EDIT: Aaand now that I've done all of it... Complete waste of time. Literally did all that and way kinkier stuff with random inmates a billion times, and we're supposed to act like Alex is teaching us something about kink. Also no benefit whatsoever after completing it. What a pointless questline.