Some basic troubleshooting ideas...
You can run the NT4 off a battery just fine. A friend of mine does it that way all the time. Try it with the battery in the NT4, set it up at home and let it run a while. Does it actually post pone the issue as you suspect?
Also, once the problem starts, what happens when you switch Left and Right cables running into the preamp? If the problem moves, that implies something with mic/cable before the pre. If the problem still stays in the same channel, it's preamp > recorder.