I've used a cheap Wal-Mart plastic shoebox with a lid to bathe mine. Or you could use something like a plastic tub that used to hold butter with a few holes punched in the top.

Also, try to keep the prolapse moist, if it dries out there's the danger of necrosis. If it doesn't go in, I don't know that vets do frog surgery. Maybe they could glue it back, but I don't know how that would work with frog skin. You could try gently(!) pushing it in with a moistened q-tip, but that's pretty risky.

Maybe getting an evaluation from a vet wouldn't be too costly, you could at least try that. You're doing everything right, I hate to be pessimistic, but I'm a bit pessimistic here.