My experience with starching is neither extensive nor with crochet items, but in the hopes that it will be helpful:
I would try dilute white glue (school glue/Elmer's/PVA glue). I stiffened acrylic felt with it, and the felt remained flexible, but was definitely better able to keep its shape.
Unfortunately, I don't have a specific dilution to recommend. I found a blog post about crochet snowflakes which says a 50-50 glue-water mixture will make crochet rock hard, so stay well below that. When I did the felt, I used just enough glue to make the water opaque - it was not a true white, just couldn't be seen through. That wasn't enough! I was able to do a second round of stiffening, though, so it's a forgiving method.
If I were doing this I would use enough glue to make the water true white, soak the hat in it (folded like in the last instructional photo of the pattern), and press out the excess. You could use towels if you immediately rinse them out afterward.
I'd want it dry enough to shape and stand it immediately like it is in the finished photo, with plastic wrap under it. A couple of times while it's drying I'd rub my fingers on the parts that are in contact with the plastic wrap - my felt showed some shiny spots on the side that was touching the plastic. Not having a drenched piece will help with that, but I'd want to remove or work in any potential glue on the surface as well.
Some side notes to close: If you're finding a lot of interesting patterns that need starching, buying a commercial stiffening product might be worthwhile - they're designed to be easier! People also use Epsom salts (supposed to make the pieces a little sparkly), sugar (similar to Epsom salts, but I wouldn't use this because it seems like a bug magnet), and hairspray (easy for a gentle hold, if you don't mind the risk of a lingering smell).
And just to say again - this is untested, but it's how I would approach it!