On a 07 BMW 328i the brake fluid is supposed to be changed every 2 years, according to the vehicle reset timer. 2K for the Mini sounds a bit often. If that Mini was mine I would install rebuilt front calipers/pads, new front flex brake lines, and bleed the whole system enough to replace all the fluid, then see how it is. I don't know about this car, but some require a manufacturer scan tool to bleed the ABS system. Prolly a trip to the googler for that answer.

I would agree with malsal that you should get a second opinion.

Did this symptom start all of a sudden, or have you noticed a change over time?

Were there any brake services performed in the recent past?

Does the pads on the front right brakes show even wear on the inside and outside pads? - the calipers should 'float' so even pressure is applied to both pads.  If the caliper is 'frozen' onto the carrier, it may have an impact on performance.

Did the shop try bleeding the brake fluid to make sure there is no air in the lines?

MINI recommend brake fluid be changed every 2k miles - was this performed?

Ken

Presuming you have a New Mini i would think there are some simple checks to do before throwing that kind of money at it, a blocked or partially blocked brake flex hose maybe.
First thing to check would be fluid to the bleeder then go back towards the master cylinder from there.
Check northamericanmotoring they cater more to the MINI owners.
Also Ken here at MM might chime in.

I've been told by my mechanic that I am getting no pressure on the right front brake.  It's not grabbing at all.  So, if I slam the brakes, it will pull to the left.  He told me that it will probably need new ABS unit with modulator and caliper (going from memory).  He quoted me an initial estimate of $3,000.  Any thoughts?