Try a KK Ultra snaffle in a 14mm width. They are double-jointed, like a French link, though not flat. I use a 14mm snaffle with my girls. Most riders think a fatter width is gentler for the horse, but as you know a fat snaffle is just too much for a horse with little room in her mouth. If you have quiet hands a 14mm is just fine.
I just saw a website where you can try bits on your horse, which is a super idea since bits can be expensive and not all will work for every horse. I do not recall where the bit-try-out site was, but Google that and you'll certainly find several places. It was a ordinary place, like Dover, not somewhere odd. Did you Google the bit you are looking for? I've never used a French link so I don't know how rare flat ones are.
ETA: I Googled "flat french link bit" and got a number of hits. Did you have trouble? What were you using for a search term?