My sister has a service dog who has cruised even on a transtlantic. The "off/on switch" for her is the harness. If the harness is on, she is working and will act accordingly. If you take the harness off, she becomes a regular dog. It's rather insane to see the difference. We did have LOTS of people ask if they could pet her. My sister mostly said yes (what dog doesn't like a little petting?) - however her dog would respond by literally just standing there. This caused no distraction to her and usually the whole interaction took no more than a few seconds of time. Had her harness been off, the petting would have been responded to by licks and jumps of joy.
We did find that if we really needed to get someplace the best bath to take was the room corridors because they were usually vacant and we wouldn't get stopped 20 times in the process.