75mph vibration

You may have more than one problem, but
Wait your slanty goes 75mph at 1950 rpm on 225/70-14s? That calculates out to a rear gear of 2.05!!
Something is not computing.Unless you have an O/D tranny. And I haven't heard of an auto with O/D for a slanty.
I believe you have a bad driveshaft. I think somebody bent it for you.
And here's why;
A tire imbalance or a bent rim, starts vibrating right from the get-go, getting worse and worse as its rpm goes up.By 75mph, that tire is doing 955 rpm.
An out of-round tire starts hammering the car up and down,almost as soon as the car gets moving.
A blown cord will wag the tail and make you seasick
But if you truly are only spinning 1950 ish, that makes perfect sense for a bent driveshaft, and you said you thought it was coming from the center of the car, and you said you saw the D/S moving up and down.
It is possible to bend or kink a driveshaft when lifting a car on one of those in-ground single-post hydraulic lifts.
Now let's revisit your 75 mph/1950 relationship.
A typical rear gear is 2.76. With your new Coopers, they would put your rpm around 2635 with the converter locked up, Allowing 10 % for slip, with cruise throttle it would still be around 2372.
So Ima guessing your speedO is generous. At least you won't get caught speeding......
If you run the car with the rear tires off, and drums on, don't forget to secure the drums with all 5 nuts.