If for no other reason than to document my notes, I'll post here what I learned after almost 3 house of Kafka-esque customer support speaking to Airmar, Lowrance, Simrad, Gemco, and finally some not-to-be-named marine parts supplier.
My boat, and I assume others of similar model and model year (2021 FSH 255 Sport E), was supplied with a transducer manufactured by Airmar, but not supported by Airmar (or apparently anyone else). It was made for Navico/Simrad, but nobody at those companies readily knows this. The model number is B148-20, and the manufacturer p/n is 000-13907-001. According to the installation instructions for this model, the hull nut p/n is 02-133-01. That is a fairly common bronze nut used by Airmar, and googling will readily show you a wrench that supposedly fits it. The p/n for the nut wrench, I hope, is 60WR-4. FWIW, you kind of need TWO wrenches: one for the nut, and one to hold the housing still while you turn the nut. Some or all of this MIGHT be accomplished with large channel locks and/or a pipe wrench, but the space down there is tight.
After 90 minutes talking to Simrad he referred me to Gemco (yet another Navico subsidiary). The brusque guy at Gemco was adamant that the wrench(es) I had found online would not work and that nobody makes one for this particular nut, even though google says it does. He did provide me some useful info -- that the nut OD is 2.88 inches -- and suggested I weld/fabricate my own wrench. I have a MIG welder and can do some decent welding. For whatever reason I'm much less capable with metal fab, but it may come to this.
After speaking to Gemco, I spoke to a marine parts supplier in California that sells both of the wrenches I hope may work (even if the Gemco guy says no). After assuring me she could figure this out and disparaging the intellect of people at Gemco and in that region of the country in general, she called me back a half hour later to tell me, "yeah, this is screwed up. I can't figure out what's going on here." What she did do is sell me the wrenches and agree to take them back if they end up not fitting (their normal policy is no returns on tools).
At some point many of us are going to have to replace our transducers, hopefully with something that is supported technically by someone and is compatible physically with the hole in our hulls as well as with our MFD's. They do go bad. I'm not sure why Yamaha chose this particular transducer for this boat, but it was not the best from a maintenance/supportability/knowledgebase point of view. I'm actually considering purchasing one of the ones still left in stock in some place to keep as a spare, but they do cost anywhere from $400 to $600.
Sorry for the rant. Been that kind of morning. Maybe this helps someone else.