The runflats have thick and stiff sidewall which contribute to the rough ride. Switching to standard tires will help.

The other issue is the lowering springs as you mentioned. The reduced suspension travel has an adverse effect on ride quality.

Another possibility is to switch to 17 inch wheels and tires. The taller sidewalls improves ride quality.

Coilovers will not help as they are performance products that improve handling as the cost of ride quality. For the height adjustable coilover, the tallest setting is still about 1 inch lower than stock.

If you install stock height springs, you can run the KONI special active shocks, these will actually improve ride quality while offering performance improvements. KONI specifies stock springs only for these shocks.

While not the same BMW, pop's 328i came with run flats, and that is just how they felt. Get some good regular tires, and road aid. modern tires just don't blow out much anymore, and if they leak you will know long before you have a flat with mandatory TPM's on the car. 19" may be too much if the tire side wall is 2" or less, pot holes in Michigan make this a bad choice.

I bought a used 2012 Mini Cooper S Countryman last year w/ Sport package. The previous owner had lowered the suspension and installed Black 18" wheels, Which looks awesome, But with the shorter stiffer suspension and Run-Flat tires the car has an incredibly rough ride.

What do I have to do to calm this thing down a bit and put standard tires on it?

My Confusion: Would stock suspension work? Are the wheels too big for non Run-Flats? What do Coilovers do? Is there a way to soften the ride while keeping the run flats?