Another 2010 N18 Cooper S owner here with some tyre stories.
Car came (second hand) with a mix of Dunlop Runflats on the front and conti sport contact 3 on the rear (non-RF). I replaced the Dunlops with matching conti 3's when it was up for it's first MOT with me, and replaced all of them with Conti 3 (non-runflat) for another two years (one rear set and twice at the front - i drive 30,000 miles a year).
2017 changed up to 18x7.5" wheels, and had to go with 215/35/18 to fit the arches with no rub - and settled on the GoodYear Eagle F1s. These were a marked improvement over the roadholding of the Contis when you really push for any extended period, especially up in the cold mountains in the south of France. Year 1 was on Conti's next year same roads with Goodyears, and with same tune, same suspension and same everything else, I was pushing much harder, with much more confidence. Only downsides, they wore VERY quickly and when it is wet, I felt it, Conti's didn't care wet or dry. Goodyears gave me about 6,000 miles on the fronts, with the rears only giving me about 10,000 miles.
Next decided to change again. For the track, I have my original R104 wheels (17x7") with Toyo TR-1s, and the daily 18" with Nankang NS-2 directional. While I was skeptical about the Nankangs due to them being HALF the price of the Goodyears, I was strangely surprised. In the wet, they aren't as confidence-inspiring, but wear rate is much better, and even those same French roads, I didn't feel any less confident than I was with the Goodyears, and still markedly better than the Contis. I'm a big fan of these Nankangs, 9,000 miles in and still have 3.5mm on the fronts and 5mm on the rear, meaning they are a quarter of the price of the Goodyears with wear factored in. Toyos... forget them. Was such a waste of money. Some BMW-driving yoik said they were the best thing since sliced bread, but I've had overheating issues and in the wet they're dangerous. So much so that the last fast driving I've done has been exclusively on the Nankangs. I may be selling my OEM 17" wheels as there's no point in them now!
Overall and from my own experience:
Best all rounder: Continental SportContact 3. (yes, for an all rounder, including wet driving, these are the best - hence being the OEM tyre for these cars - but are very expensive)
Best fast road tyre: GoodYear Eagle F1 (expensive, but worth it if money isn't an issue)
Best Summer tyre for daily driver: Nankang NS-2 (as good as the Goodyear day to day and when occasionally pushed, but way better wear rates)
Would avoid: Toyo Proxes TR-1 (don't listen to yoiks..)
Hope this helps someone.