I am willing to bet that the transformer itself is the reason for the ABS code.

Neon transformers create a lot of voltage, as such, that creates what is called "EMI" - Electro-Magnetic Interference. If it's a cheap ass transformer (which I suspect) and poorly grounded or isolated, all hell breaks loose. That's why your ignition is not the cause of the radio quitting.

The shield your dad made is not a bad idea, but really, a waste of time. It's masking whatever is really wrong with the install. And judging from the wiring job(s) done on that car, I am surprised it hasn't melted down. Seriously.