Excellent news, comrades ...
The transformer was supposed to arrive yesterday; but then UPS said the package was damaged and they're returning it to seller. So I asked the seller to tell UPS to deliver it anyhow and let me inspect it. Apparently they told seller they couldn't find it, so more or less simultaneously today the transformer and a full refund arrived. The transformer was indeed damaged. So I told the seller what's up. I dunno who ate the cost: the box was badly damaged, so maybe UPS. If it was the seller, I'll offer to pay half-price; otherwise I'll consider it a gift from UPS

Anyhow ...
View attachment 347086
I hooked it up to a 120vac/20amp duplex outlet (measuring 122vac) and to the 240vac Romex going out to the pressure switch&tank and the pump (measuring 238vac with the pump off). When the pump came on, the two sides sagged to 119vac and 228vac, respectively. Everything seemed to be working fine, drawing about 9 amps from the 120vac side. I'm ok with the voltage sag, given the pump's spec is 200-240vac. When the pump was off, the transformer drew 1.6 amps; this 200 watts is unacceptable waste, so any scheme that leaves the transformer continuously energized is out. The 60Hz hum is very mild though, not an issue. I unplugged it while the pressure switch was "on" and plugged it back in; it did not pop the breaker, but there was a noticeable "thump", so not liking that scheme either. I then connected the 120vac side to a 1000watt 12vdc inverter which I can connect to our Nissan Leaf EV (which is my current inadequate backup scheme), and that worked fine too; so that gives me confidence a 120vac power station with at least 2000watts output (Ecoflow Delta 3 Max or the like) can handle it and a few other loads.
So I think I'm going to stick with the scheme I drew in post #59 above.
Also, how the heck to you ground this thing ? There appear to be no ground terminals on it. I'm going to put it inside a NEMA box (along with the DPDT relay and the load switch of
@Whitenuckler ) and that will be grounded, so just the mechanical connection of mounting it to the box ?