...that a Chevy Cobalt compact spare tire will fit my 2012 cooper S ? I want to ditch the run flats .

 

You don't need a spare to get rid of the runflats. At a minimum you need a compressor and a can of slime (not fix-a-flat). Some people also like to have a plug kit or AAA card and cellphone.

 

That's what I have.

 

...but I have a tendency to pick up nails/screws in the edge . I feel better with a spare when I'm 30 miles from the nearest town .

 

Two things that needs to be checked are: OFFSET and HUB BORE.
The Chevrolet Cobalt has two different bolt patterns, a 4x100mm and 5x110mm (5x4.33") lug bolt pattern; the MINI is a 4x100mm.
By the time you search for a working non-MINI spare, you'll still need to put the correct tire on it.
Might as well grab the correct spare wheel from the dealer or MINI parts supplier and slap a correct tire on it.

- Erik

 

Honda civic 15" space saver from wreckers. If you want to be sure ask if you can test fit it. Job done and money saved!