Twine has several flavors. Two of them would allow for deep customization of the UI, the Harlowe variant does not really allow for it without some serious HTML skills. I've seen some really good Twine custom UI's done but they were from skilled developers.
You can also NOT use an engine at all and just use standard programming skills, C++, Java, Perl, hell, even Visual Basic can make custom UI text games.
So really, the question you should be asking is what skills are you bringing to the table and then you can properly choose a programming platform to go with it. You'll likely be facing a developmental learning curve no matter what you do unless you have some skills already handy.
The earliest versions of games like this could even be done in WORD or just standard HTML. I saw a similar CYOA game done in PowerPoint... a crap-ton of work but it looked slick and just bounced between slides and it was in .EXE format which PowerPoint can export.
So, my advice is look first to your existing skill set, figure out how much you are willing to learn on your own (aka: how steep of a learning curve can you handle) and then use that information on how to narrow the field of choices.
BTW: Mythos has a now classic adult game (Fools Quest) that is written in a early form of software that became Twine with a custom map. It could easily be done in current Twine but I think the version used is an early custom version that pre-dates Twine 1.
P.S. You can also 'wrap' any standard Twine variant game inside of an HTML UI wrapper using the iframe or other insert methodologies (there are a few). So, your engine would just the created text and send out HTML commands to the wrapper to change the UI if the game itself needs to have the UI change. I'm providing this if you are versed enough in Twine and just have some cursory HTML skills. It's not a far step from that point to a game like you describe, but you'd need to host it someplace. You could even do it in WordPress but they have some rules about adult materials on their site... so you might look for another option.