When the sound is turned to moderate or high volume, the amp goes into "safety mode" after a minute or so of playing,

The amplifier goes into safety mode whenever there is too much current draw on any one (or more) channels. It isn't because the amplifier is overheating.

Too much current draw is most likely caused by a bad speaker wire, where the pos (+) current is being shorted to ground. Reversing your phasing (I assume by swapping the leads at the speaker) would not change this.

It is also possible that one of the speakers is defective, with a shorted voice coil, although this would likely manifest itself with poor quality (or no) sound.

Test by disconnecting the speakers one at a time, and seeing which channel is causing the amplifier to go into safety mode. Then replace the wires to that speaker, and if the problem persists, then replace the speaker.

tanstaafl.