I'm going to grab the low hanging fruit here and say try supports. Just because you shouldn't have to use them doesn't mean you don't.

And saying People use supports like a crutch seems quite elitist to me. There are many different ways to skin a cat, and if it works, it works. If this was a production/commercial setting, getting the initial run to work using supports, management wouldn't care, as you had a working product. If you then refined the process to not use supports, they would be even happier as you are now saving them money. But wasting time delaying production on the perfectly optimized setup never goes over well. And since you're obviously not doing this for a commercial run, copyright laws and all (and if you were, you would be using injection molding, not 3d printing, given the size of a run you would need to turn a profit), I don't see what the hangup is over supports. Yes, they're a pain to clean up sometimes, and I try to avoid using them whenever I can, but If I need supports, I use them. One of my commercial products I do print I have to use supports, and I do, and I spend the 20 minutes cleaning up the piece.