My last post was pretty long, so I left out this part. I found a spare bit hanging up in the tack room, and Cathy said we could try it. It looks like this, though I'm not sure it's brand-name. We were using a eggbutt french link of uncertain maker that fairly closely resembled this.
Izzy definitely appreciated our switch from single jointed snaffle to double jointed that we made several months ago. She's pretty sensitive about her mouth, and she's much more willing to go forward into contact with when the nutcracker action of a single-jointed bit was gone.
The bit we were trying was loose ring, which I am leery of. I know they're supposed to be great, but they can pinch the horse's mouth and my old mare had several bitting quirks, which meant that when I put her in a loose ring, she picked up a whole new vice (clicking her teeth). Izzy isn't nearly as fussy with her mouth, so I tried the new bit and tried not to hold my breath. I did however, firmly resolve to dismount and unbridle immediately if any teeth-clicking occurred. That drives me nuts.
Izzy went surprisingly well. She did flip her head a little bit, but I think that was partly because she wanted to be done working and partly because it was adjusted a little lower in her mouth, which let it bounce around. She also braced against me less, which is theoretically a benefit of a loose ring bit, but she usually only braces when I'm stiff and just letting her go around. I was actively making her work yesterday, so it wasn't a situation in which she'd usually brace anyways.
We'll try it again tomorrow, and see how it goes.