There's no two ways about this, a faulty monitor cannot cause a PCB to start resetting like a good-un. It just can't. Even with a very, very slim possibility of a tiny, bit of feedback on the X / Y / Z outputs of the PCB, that's still only the analogue output section of the PCB (DACs etc.). It
could impact the PSU and in turn the AR2 though.
My money would still be on power, or PCB. Somethings failing when it's getting hot and overloaded. When Mark tested it, was he running it on your Atari power brick and AR2? I guess not, or even on ANY Atari power brick and AR2. Again, knowing Mark (and I'm not at all saying this in a bad way!), I doubt it - I suspect he's built his own supply voltages for his rig.
I don't know where to tell you to go from here, but it would be interesting to rig a meter up to the 5v from the AR2 and see what it's at to start with, keeping monitoring it and seeing what it's up to at the point the PCB starts resetting.
I know you don't have one of these but at this point I'd be looking to stick a couple of the PCB voltages on my new wazzy scope that can give me max VPP and min VPP and see if the voltage(s) drop too low at any point. But I appreciate that you can't really do that without the kit to do so