Her hair is saran, which is very durable but needs heat in order to properly set. Boil perming is really the safest method that will work on it and have the least risk of the fibers melting. As coinoperatedgirl said, it's very easy and a lot less scary than it sounds.
I was really hesitant at first before trying it myself, but I've boil-permed handfuls of dolls in the past three years, some repeatedly to get the results I'm looking for, and I've yet to harm any of them in the process. Make sure they're nude when trying this however, the colors in some of their outfits may bleed if they get wet.
I section off the hair and use drinking straws (milkshake straws work best for larger barreled curls) fastened with bobby pins for curlers. Just set a pot on the stove and let it come to a boil. Pour the water in a ceramic bowl and dunk the doll's head for about 30 seconds, then remove and place in ice water for a few minutes to set. You may want to avoid dunking the face in the hot water, as I've heard of some problems arising with the paint from doing so, though I've dunked some entirely with no issue. If it happens, just don't touch the paint with your fingers until the plastic cools. I usually let the doll sit for 48 hours, or until completely dry, before removing the curlers. If the fibers are still wet then the curls will fall down.
The only dolls I would be mindful of are those that have flocking (Venus and Clawd) because it will come off in water, or any dolls with kanekalon hair (Jackson, Venus, Ghoulia, Robecca, Spectra, and the pink in Draculaura's hair). Kanekalon can still be boil-permed, it's just a little more sensitive to heat than saran, and may frizz.
You can also try alcohol-free, water based hair gels (essentially what Mattel uses), or hairspray intended for wigs, to set Operetta's victory rolls, but then you'd just end up with stiff hair again.