More (worthless?) info:
Engine is now probably about 4 years old, maybe 3? Less than 500 miles.
Assembled by a pro shop that specializes in Mopar and race engines.

Problem is that, since the build, the car sits for 6 months at a time. Although, I've been trying to rectify that, though fighting the issues is getting it close to being mothballed!

The timing was originally set, and unchanged (until the weekend).
The balancer was a budget brand, known to have failures, so I replaced it. Unless the chain slipped while replacing the balancer, the timing should be unchanged. Indeed, the weekend was the first time I put the timing light on the damn thing. It was showing 20 BTDC, so I changed it to 16 BTDC to see if that helped.

If I've got time this weekend, I'm rebuilding the carb and hopefully toss in a piston stop to verify TDC.