So, that was it. I removed the burned out fuse and took it to Lowes. It was labeled T3.15AL250V, which I take to mean it is a 3A fuse. Lowes only had 5amp fuses in that size (5x20mm), so I got one, understanding the risk of going with a higher fuse.
I installed it, with power to indoor and outdoor units off, and to be safe, cut 3 inches off the cable and stripped new wire, installed all, turned power back on and no display.
It's utterly miserable in the attic right now, so not sure I'll give it another go tonight, but 1) if it is blown again, what could this be (power was off the whole time this round), and if it is not blown again, what would be the next logical thought, damage to the board?