They're right that the best way to solve it would be to run new 12-2, is that feasible? Crawlspace/attic access?
What I've seen since I've moved to backwater is exactly what your neighbor mentioned doing. Here's the fun part right. Without rewiring you could just run some raw cables into your house and try what your neighbor suggested. You're not going to hurt anything everything has breakers that will prevent overcurrent conditions before anything bad happens. This assumes the genset people did their work right. If they told you about this problem instead of just "setting it up" they're probably descent enough. If you're that concerned you could get an IR camera or use a thermocouple on the lines to see if the wires/breakers heat up. That's the only reason you have individual breakers is to save your *** if you pull too much current through the wire, the ways of doing this are of course innumerable.
Personally I would much prefer if you fixed it the right way as will the next guy but, I'm getting a little over my hope to fix everything properly, as long as it works safely. Of course leaving notes/pictures of what you did that deviates from code would be really nice but also probably incriminating. After all NEC is there for 2 reasons, to stop you from killing yourself, and stop you from killing the next guy who owns the house.
You seem pretty descent and I wouldn't fault ya much for doing it either way. The next guy will be glad to have the GenSet regardless.