Something in your wiring is failing when you generate a certain amount of heat in your set-up. I had a plug do this on me before I upgraded everything on it. Bypassed the plug and wired it direct and problem solved.
Get your hands on a basic multi-meter. Those things are invaluable for so many reasons. Check resistance levels throughout your system and isolate the plug. Check the wiring...then check the plug by itself. Also check voltage levels before the plug and in the plug.
The other possibilities are that you have a weak wire somewhere in your run which exposes itself when heat is built up causing it to cut out. Of course make darn sure your connections at the battery are rock solid.
If it is in the wiring you might want to consider re-wiring the thing and if you do step up a size in wires. If they are currently building up heat then by going a size bigger you'll have more surface area on your wiring to dissipate the heat.
I'd still get those batteries load tested just to eliminate them as a culprit. Most auto parts will load test batteries for free or you can pick up a load tester at a Fleet Farm for around 30 bucks. While not as invaluable as a multi-meter, having your own load tester if you have a bunch of vehicles around is a good investment. My parents have a farm and the day I bought a load tester is now looked upon as one of the best tools anyone has ever purchased.