My brother pointed out to me that one thing we didn't do was check the pickup/reluctor gap on the distributor before installing it, so that is something we are going to check next time we get to work on the car. But could that possibly be causing this issue? Would low power to the ECU cause it to be more sensitive to there being too wide of a gap?
Regarding some of the other suggestions, we did remove the bulkhead connectors and sprayed them generously with deoxit (they were kind of cruddy) but that didn't make much, if any, difference.
I am tempted to just install 2 relays to feed 12v directly from the battery to the ECU (w/ a fusable link) and connect the coil from one to the Run and the other to the Start. I know I wouldn't be actually figuring out what the root issue is, but I'd feel better about the ECU not having to get power through the BR during cranking. Seems like that'd be a better setup in general.
As for just removing the BR only if the coil has internal resistance, this is the type of coil on the engine:
ACCEL 8140C Ignition Coil - Chrome - 42000v 1.4 ohm primary - Points - good up to 6500 RPM
and these are the specs:
Ohms Secondary Resistance 9.2 k
Ohms Turns Ratio 94:1
Primary Resistance 1.4
I don't know if that's what you were referring to though by saying "internal resistance"