Is it really not feasible to wear road shorts in the water and run in them provided the chamois is thin enough to run in?
It is absolutely feasible. I have raced in a few tris with a pair of road shorts. The road shorts I use for tris has a thinner chamois as you suggest. The best thing to do is take a pair of your road shorts and go for a run and see how they feel. If it feels like your running bold legged or running with a **** in your shorts, then you should try a different pair
It's probably also a good idea to take your bike shorts and go for a swim with them. A word of caution though, I wouldn't take your most expensive bike shorts in the pool, as the chlorine might not be that good for them.