New facts, for those following along and for those who might have some insight:
1. RP light does NOT come on when using the inverter or shore power. So, the problem is when getting power from the genset only.
2. Using a plug-in polarity tester, I checked the polarity of all the outlets when under genset, shore power and inverter, and they all are fine (polarity not reversed, and neither ground nor neutral is open).
3. The individual circuits turn off and power off when I throw the breakers for them (outlets, water heater, AC and battery charger), so I really don't think the problem is a reversed wire somewhere.
4. The RP light, when on, is very faint and you can't really discern it in daylight unless you cup your hands over it. Um,

(sheepish grin), it is possible that this has been going on since we took delivery of the boat and we just didn't notice because it's so hard to see. I noticed it yesterday because we were using the generator after dark, which we normally don't do.
From the above, I conclude that either (1) I have no problem (some learned folks have written that a faint RP light may not be an indicator of a problem depending on how the charging source is wired), or (2) I have a loose or bad neutral or ground wire somewhere in the genset line between the switch choosing between shore/off/gen and the generator itself. I've looked and I can't find anything, but I don't feel all that comfortable dealing with A/C issues, so I'm going to have an electrician check it out Monday morning. I always wanted to spend a few days in Huntington anyway.