Glad you're gonna save it. An early RCA with a newer CRT is a recipe for an excellent picture.
Please try what the others suggested and disconnect it. Test it on the bench. The forums over on antiqueradios.com are full of old threads where someone was convinced the transformer was shorted, and the problem disappeared when the transformer was totally disconnected. Shorted tube sockets, dial lamp sockets, and such. I'll admit what you said sounds really bad for the transformer, but humor us and try it. If it's bad, it's gotta come out anyway.
Even if it's bad you still might fix it by replacing the insulation under the end bells like Chad suggested.
I don't have a crank megger. I use a HV power supply and a VTVM. You don't happen to have these two items, do you? I suspect this is doing exactly the same thing as your megger anyway. If you have this stuff laying around, post back, I'll tell you how.
Ok, if it *is* toast, I don't know of anyone rewinding, but I've heard Heyboer will make you new transformers to original size and specs. This might be a good option if a universal one wont do for some reason, or you want an exact fit.
Good luck,
John