Cuda won't start and have tried multiple avenues. Need help!

Recently I've run in to troubles with my 1970, 340 Plymouth Cuda 4 speed. I go to start the car and the dome light and parking brake light will come on, but when I go to start the car, a single click happens and all the lights go out. When I try it again immediately after, no lights come on and the car is unresponsive.

If I give the car several hours or even a day, when I go to start the car, the lights will come on again, but it follows the same process as previously stated.

I have replaced the starter relay twice within this last month, and I've even gone as far as replacing the starter itself. I believed that the starter was failing, causing more amps to be pulled through the relay and exceeding the relay's amperage rating, causing it to fry the internals. After replacing the starter motor and still having the same problem makes me believe it is not the starter causing me problems.

When the lights go out, I test the voltage in the battery and read right around 12.6/12.7 every time. When i test the relay to see if it gives me the same voltage reading, it shows right around 0.125-0.200 volts.

At this point I have no idea what could be causing tis issue. I don't know if maybe the neutral safety switch is going bad, or if there is a wring issue between the two. I wasn't the one who restored the car, but I bought it from the previous owner who did the restoration. The car has lots of original/period correct parts on it.

What could possibly be causing me these issues? Am I overlooking an obvious issue?