Mark, is the kicker wired off the House battery? I am assuming your electronics are rigged off the House battery? Were you observing voltage on your electronics?
If so, you probably saw your House performance only, via the kicker. If so, that sounds normal... using voltage as a metric, 12.7 volts is ideal for a new battery in good shape that is fully charged, with no load, measured while "resting".
Slightly less is ok for an older battery (ie, 12.4, which equates to about a 75% charge in a 12 volt battery). Less than that in a fully charged battery that has been sitting overnight with no load on it means the battery is weak and should be replaced, or "restored" with a smart charger (may or may not work). When charging a depleted battery via a charger or your engine's alternator, it can go up to 14.4 volts. Once a battery is "fully" charged, it should drop to around 13.4 when the charger/alternator switches over to a maintain or float mode (if it keeps charging at 14.4 it will cook the battery).
If you saw it go to 13.5v with the kicker, that likely means the kicker charging circuit was working fine and the battery was at 100%, with the kicker putting out a little extra charge to offset the load from the electronics.
You wouldn't see anything about the main engines charging in this scenario unless you ran the main and observed the ACR kick in and start sending some charge to the house battery, or you could switch your electronics over to the Start battery and observe it. Another option would be a cheap little voltage meter on the Start circuit, or a clamp on meter like suggested above. If your Yamaha has digital gauges, depending on how the dealer wired them it may also be possible to pull up Start circuit voltage, or even observe alternator output on your engine (my Honda gauges do this).
In any event, it sounds like your kicker is charging just fine, but may not put out enough charge on it's own to offset the draw of your pot puller. Remember the kicker's rated alternator output is typically at WOT.... at idle it puts out a fraction of that. I believe the Yami 9.9's put out 6 amps at WOT, probably 1.5 at idle... that basically equates to a trickle charger at idle, and a basic home charger at 6 amps. If you fully depleted a deep cycle battery, @6 amps you'd be looking at 5+ hours to fully recharge it. Basically, your kicker has no hope of keeping up with the drain of your electric pot puller.
I generally think the T9.9 are the cat's meow on kickers, but one area the Honda 9.9 high thrust shines is the alternator... 12 amp output vs. 6.
So yeah, I think using the main is your best option. Just mix up some extended periods of hard running with the long periods of idling.