Looks to me like the clicking sound and the ammeter twitching happen at the same time? I have no guess on the clicking.

The ammeter does not use the voltage regulator like the fuel gauge does. I noticed the fuel gauge didn't come up any with the key in run. 3 gallons should get the needle to E.

I think you need to find the clicking source first. Is it under the hood or definitely under dash?
You should get the dash frame grounded good too.

If you momentarily ground the fuel sending unit wire, the fuel gauge should peg upscale. Or try grounding the temperature sender wire or the oil pressure wire. All 3 should peg upscale when grounded.
BUT BUT BUT only ground them briefly to see needle deflection otherwise gauge damage could result.