Behavioral therapists are very good at solving problems like this. You might call one and ask how many visits would be involved, what would it cost, and figure out the net cost to your wallet after insurance, co-pays, FSA accounts, etc. If it was me, I'd do it. (Get a recommendation from your pediatrician.)
If it would cost you net $180., for example, ask yourself, "Is it worth $180. for my son to have peace again regarding bath/shower?" The answer will probably be, "Hell yeah!"
And you're totally right about the younger son learning this behavior. My kids ARE NOT ALLOWED to express dislike of any foods at dinner, because the younger ones pick that up in a heartbeat and then also refuse to eat that food. They're very impressionable. (I don't force my kid to eat that food item; I just tell them they're not allowed to discuss it, frown at it, or in any way let their fellow diners know they're displeased.)