Choice
The main issue I see with this is that the choices won't matter. If it literally just ends up being superficial level differences, I think you'd be better of choosing what monster is in the story and fleshing it out to create the best experience that way. Granted, I think there are some interesting ways this can be handled for a horror story, so it might be doable even if the customisation choices don't matter, but I get the feeling that isn't what you are going for.
Now, if the story gimmick is that there can be different monsters and that it actually matters, then the issue is the amount of work that'll require. I'm not sure how you are planning to handle this, but keeping the scope of the story in mind would be a good idea. Having three different monsters vs hundreds of combinations (i.e. claws & wings, claws & no wings etc) are very different beasts. Further, branching stories can get out of hand very easily, which is why it is often recommended to start small.
Advice
If you haven't written branching stories before, I'd advise picking something with a clear, limited scope to work on first, to get some experience, both with the process (to help you find what works best for you) and with the editor (so you can actually put it together). Then you can try tackling this more challenging idea later when you have a better idea of what to expect.
Obviously, if you still want to do this idea, that is fine, but I would advise planning it out a bit to figure out the specifics for how you plan on handling this.
Idea
Almost anything can be interesting if written well, so the idea can certainly work. However, personally, I consider writing horror a bit harder than something like generic fantasy, and adding in customisable monsters seems like something that could lead to important scenes lacking tension (but this does heavily depend on how you handle the monsters differences in the story).
What you mention in your post can be done, but I question what comes after. Regardless, write what you want to write, because the worst case scenario is that you learn a lot (even if the story doesn't pan out).
Recommendations for Improvement
These are a bit hard to give without seeing some of the actual story's writing, but if you follow general conventions and writing rules you ought to be fine (as deviating from them better be done with good reason). Things like:
Obviously, more specific advice could prove a lot more helpful, so do tell me when you write some of the actual story if you'd like me to take a look and give less general recommendations.
Conclusion
Idea can work, however it being ambitious makes me worry it might be difficult to execute, but if you keep the scope in mind and don't let it get away from you I imagine you ought to be fine. Write what you want to write and all that, because when the worst case scenario is improvement, you might as well try.
The best way to improve the reader's experience is to write well, but that is such a broad statement that I'd say ask this question again when you have some writing to show (so that you can get more relevant feedback).
Go write!