Had a similar vibration on my '60 Tbird but not at take off.
Tried a bunch of the same things - had the drive shaft balanced, u - joints, tires, rear wheel bearings.
Believe it or not - mine mostly went away when I recently replaced the rear shocks (went with a heavy duty style). I think the small vibration that is left on my car is in the rear transmission bushing since it happens mostly at higher speeds. Don't the '61's still have the Cruise-O-Matic? Might need to look at that rear bushing. Not sure on your diff compatibility - hope the other folks here know.
Good luck - let us know what you find