Grounded Tach Wire Creates a Gremlin?

For starters you have nearly .4V drop in to the VR. You said you measured battery at 11.95 (that is low) and VR at 11.36. That is not a lot but it adds up.

I don't understand your grounding description, is this trunk mount battery? That is cause for concern voltage drop

Check the VR GROUND Whether or not trunk mount, "rig" a wire (does not have to be large, no.18 is big enough)--- long enough to reach from battery neg post wherever it is, up to any part of the car. "Rig" the voltmeter to read from battery NEG post to the VR mounting flange. Run engine at fast idle "low to medium cruise" and check with loads off, and again with everything powered--lights, heater, whatever you have. You want a VERY low reading, lower the better, zero is perfect

Possible causes IN NO ORDER
1...Get a known GOOD and CHARGED battery in place for testing Battery can be sulfated. Rare for battery to cause problems but they can
2...Also rare on solid state VR is they can be off on voltage. But it can happen. Check voltage drop as mentioned first, both in the ground path and the blue wire "IGN" terminal of the VR all the way to the battery
3...Probably the largest single cause of over charge in Mopars is voltage drop to the VR both hot side and ground side

I don't see that your tach wire could have caused this, by the way