Sector 111 is also known in the MINI and Solstice/Sky world as Kappasphere. They have an R56 so it may behoove a call to see if they have tried to fit them.
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The rear wheels from the Elise/Exige will fit on the MINI, however plan on modifying your wheel arches to clear the tires.
The wheel dimensions for the sport package Elise or base Exige [the double-7 spoke forged ones] are 17x7.5 et38. I'm not sure S111 will sell you a set with 2 pair of rears, but you can ask!
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Basically what I figured. 215/40 would probably be a good choice for a street MINI with 225 for the track with the arches WELL modified HA HA HA. My setup pretty much tore all the plastic off on their own, still rubs with big bumps though. Some time will wear it down nicely HA HA HA.