My guess is something about the waveform of the inverter is tripping up something in the power supply. Many moons ago I worked on laser printers and we had a customer who was melting fusers. Turned out their UPS had small but fast spikes around the zero-crossing that would exceed the allowable dV/dt and trigger the triac to turn on even though there was no gate signal present. That's not the mechanism here but I would scope the AC line from the inverter. I bet it's not clean and something about the weird waveform is confusing the power supply. I don't think the 4500 W inverter is even noticing the loads from the router power supply. The power supply is misbehaving browning out briefly. Nothing in the router will run straight from that supply though, there are DC-DC converters to make the lower voltage used on the board. It would be kinda fascinating to figure out what is really happening to lead to the described behaviors.