Thanks for the advice all!
I rode a few miles this morning with the gas cap open. Still having the same problem, so that's one thing to cross off the list of possible causes.
The bike is a 79 special, almost 100% stock, only 8,000 miles. Still using the points ignition. I bought the bike from a work buddy who redid the points and timing before selling it to me. The only thing not stock on the bike are the carbs, which are the PWKs from Mike's. The battery is new, but I haven't load tested yet.
Carb jets are clean, but the carbs haven't been properly synced and they are acting like there is a vacuum leak (RPMs stay high for 10 or so seconds after reving). I know the carbs need some work, but since it's running fine at first I don't see how the carbs would be responsible for my troubles.
On the charging system.. I have noticed that headlight brightness corresponds to RPMs, low RPMs=dim, high RPMs=bright. Not sure if that tells us anything? I've never had to use the trickle charger and the battery seems to keep at 12V. However, I haven't tested that battery while the bike was running. Also I haven't checked the brush length.
Seems like the next step would be for me to check the voltage while the bike is running, and hot, since that's when my problem shows up. Will do that and report back. Again thanks for the advice!
-Jake