Charlie,
the only solution to the LEDs lighting up during SSB HF transmissions, is to tack a .1ufd capacitor across each one. hardly worth it, and as someone once noted, the LEDs make for a good RF output indicator. that is to say, don't worry about it. yes, just ground one end of the tuner control cable. it's an RF consideration and not for noise. many boats have RF feedback problems, RF getting into computers that require these sorts of measures to allow Winlink/Sailmail to work correctly. think ferrites!
I've tested many antennas, balanced antennas, without ground and they work considerably better than the backstay. though I must say, the isolated backstay works surprisingly well with a tuner. there are just too many shrouds and spreaders to make the other implementations practical at anchor. on my 42' HR, I couldn't quite string the full 102' version of the G5RV, but the shortened design worked very well. I'd be happy to discuss grounding schemes in another thread, though you can hardly improve on yours. you might try copper braided strap to a thru-hull and compare results. I doubt if you'll find much of a difference without using instrumentation.
for grins, I tied wire to a hydrogen balloon and launched it about 200' as I recall. worked fantastic, and as luck would have it, one of the contacts on 75 meters was a ham on his steel trawler on the Columbia River, with a helium filled balloon antenna. slim odds!
it's clearing up and we're off sailing. I'll listen for KC1S on 3870 Khz in the near future. you lucked out to get that vanity callsign. no way is it just a coincidence!
enjoy the weekend ...
daniel