I've been through this as well, took two years of figuring out what it was..balancing, new wheels, new tires....first off.

1) what year/model MINI do you have ?
2) how many miles on the odometer ?

And with respect to OP's need to purchased hubcentric rings, i get mine on ebay.

 

Could possibly be an internal fault in a tire. To check, run pressures in all four up to 50 psi. Drive car at speed that vibration was worst on level, smooth road. If vibration is gone, drop pressure to 36 psi on one tire at a time and drive again. When the vibration returns, you've found the faulty tire.

 

I got thinking about this more, at my age keeping the brain going is always good. Now perhaps some race engineer will chime in here, but I am wondering what the impact is on the rotating force of the wheel/tire as the tire is compressed when it contacts the road and the weight is applied. Symmetry is impacted and IMO this would have an impact on the rotation force.

Back to the "hubcentric"/"lug centric" discussion, with the lugs tightened correctly and even with a small amount of tolerance, in this instance thousandths of an inch, I am thinking the flexibility of the tire would compensate for any "x" axis wobble.

This same thought process would appear to apply towards NCTrackRats suggestion of increasing inflation to 50 PSI. Would think this makes the tire more rigid that then decreases the impact of tire imbalance on the "y" axis wobble.

Oh, and yes I have even thought about the "Chicken and the Egg" and have an answer for that as well.