I've had my 2005 Mini JCW for abour 2 years now and have run into a tire size problem before when one blew out..

Anyway:
I have four 205/55R17 tires and am looking to replace only two of them due to money constraints. I feel like these tires are too big (came with the car) and occasionally rub when making right turns out of driveways. What size would be suitable to swap them out for and would it be possible to drive on different tires for a few months until I can buy the other two?

 

Yes, those are pretty tall, 205/40 or 45R17's (I can't remember which was stock) would be best. You could probably get by with only 2 for a while with no big issues, just put the new, shorter ones up front for better steering clearance.
Honestly, the only issues I've ever had with running different sizes is with widths, the shorter and lighter the car, the more issues you'll have with different widths, when it comes to how they track and deal with grooves and irregularities in the pavement.

 

I would think that tires of differing diameters might cause problems with the traction control and other systems thinking the shorter tires are spinning or underinflated. 857 (205/45-17) revs per mile vs. 802 is almost 7% faster spin rate.

 

I would change all 4 to 205/45/17 or 215/45/17. 205/55/17 is pretty darn tall for the wheelwell.