Biggest tire I can run on a 1964 plymouth valiant

I sure don't, pretty sure they don't make one. It's the size you need to maximize your tire width, but like I said a 15x7" with a 4.5" backspace is not a standard Mopar size. If you want something you can put a dog dish on in a 15x7" with a 4.5" backspace you're probably in custom wheel territory.
Good morning Brotha

Thank you again for that awesome message, by far has been the most detailed answer that has helped me so much.
I spent the morning calling custom wheel places and the 63 series in big bolt pattern 15x7 4.5 BS even with custom wheel guys is hard to find lol.

My other option was going to be a 235/60/15 on the 15 x 7 4" BS rims I have access too. Do you think that would work and also allow the car to sit at stock height with the wheel nice and snug into my wheel well ?