After a bit of probing, it became clear that the RFI was entering on the internal home phone wiring. Disconnecting it from the RG quieted the problem, but alas we are now very dependent on the U-verse VOIP service. Disconnection is not an option, normally. (Cell coverage is poor here - a story for another day.)
The AT&T installer had not fully appreciated my phone entrance system, which uses a special DSL surge suppressor that is bolted onto my Single-Point Ground system. He ran his CAT5 cable from the wall plate straight into the RG. (That's OK for RFI, but not great for lightning surges.) The house phones were fed via a circuitous route passing (backwards!) through the surge supressor, and generally making a nice RF pickup loop.
This afternoon, I pulled out half my station (the heavy part!) so I could get access, and rewired the phone system, adding ferrite chokes, etc. Result: no trouble now with 800 W at 3600 kHz. I'll do more tests later, when the contest goes away.
U-verse RFI tip: You can use the ubiquitous DSL telephone line filters as an RFI choke. These are transparent to telephone audio, but block the DSL frequencies. (It's not clear whether they are effective at up to ~8 MHz that VDSL = U-verse uses, but it seems likely they are.)
More complete info will be at http://aa6e.net/wiki/RF_Compatibility.