This is an eternal struggle.

8 volt batteries are a crutch for tired, old cars that have not been kept up. Look at the battery cables, make sure they haven't been replaced with auto parts store 12V cables.

I’m just not a fan of these on any ‘56 but a Caribbean. It crosses a line for me. I love seeing the wire wheel-look hubcaps. The wire wheels were in the accessory catalog, technically you could have them. Doesn’t mean you should have them.

Single tone paint jobs weren’t that rare in 1956, they were usually later build dates than early. My former 400 was single tone, all Norwegian Forest Green.

Just get a new button assembly. The world is full of them. Not hard to get.

In addition to Howard's always excellent advice re: the electrical operation I am wondering if there isn't some mechanical binding occurring that may be causing this start-stop behavior? It would be beneficial to get the car on a hoist with treads you drive up on, not the type where the suspension hangs as the car is supported by the frame. Have someone up top load (work) the system while you watch the bars move.

Look for an old industrial building used for car storage, especially if there's a ramp up upper floors. That for me is the best affordable car storage. The worst: Pole barns in the country. An open invitation to moisture and mice.

I can help you there!

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I remake the whole finger and lead harness, all soldered connections, no dry crimps anywhere. They are only available as part of a comprehensive rebuild of the pushbutton actuator. All brand new wiring, no relying on 66 year old wiring and joints for reliability. Unless the customer asks specifically I make the finger harness in the "automatic park bypass" way. The Park finger has the same number of connections as the other 5 gear positions. This gives the best reliability.

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Just bypass the auto park relay per the service counselor. That feature is just trouble. I hope your terminals are soldered on to the wires and not just dry-crimped.