The issue with the Ford design is that the studs will hit the flywheel, I have never had to pull a converter stud out of a Ford, but in theory, if you could, you shouldn't have to back the tranny up. The pilot would be close, but I think it would work. The issue would be keeping it from sliding forward again as you revved the engine. If it did, it would spin up and being unbolted likely make scary noises as you are listening for less scary noises LOL Not to mention, you still won't really know the cause, just maybe the location.
I would have to guess, unbalanced converter, wrong converter pilot size, or bent/broken/wrong/poorly seated flexplate. Although I do agree you could fire it, I am not sure you are going to get anywhere with the test
I'd likely slide the tranny back and drop the flywheel and pull the converter and see what you can see.