I think it sounds like a worthy experiment. I have never had this plant, but I thought I would chime in just to get things going on this. The native Mexican hats around here are much shorter, but they are still kind of floppy guys. I think the general rule is that if you do early summer pruning, then the plants will bloom a little later than they would if they weren't pruned. Also, there probably isn't guarantee that still wouldn't need to be staked.

I leave them to self-sow in my native gardens. I have never cut them down or handled them in any way. Mine stay upright. They are in full sun and have never had extra irrigation. They self sow to abundance.
Leaner soil and less water should keep them upright.

Given the number of yellow, daisy type things out there, I'd replace them with something that is going to be naturally shorter. Heliopsis maybe. Keeping these at 3 ft sounds like it is going to be a challenge.