Why won’t my Champion C46540 generator run the microwave?

You should have 4 wires running from your generator to your panel. I'm going to guess at the colors, L1-black, L2-red, neutral-white, ground-bare. The AVR regulates L1. My guess if your microwave is on L2.

Swap the red and black wires at the panel. This will put the microwave on L1. The problem is this will move other loads, maybe the coffee pot to L2. The coffee pot is a lower resistive load so it should have less affect on L2.