Does anyone know what brand of paint BMW uses to paint Minis?
The MINI factory paint is a water-based, extremely-low VOC mixture designed to be applied with factory equipment and then oven-baked. I don't know if independent shops can get it, or even if they'd want to use it if they could.
Any of the major auto paint manufacturers (PPG, Sherwin-Williams, DuPont, etcetera) will be able to provide paint that matches the factory paint perfectly.
I'd suggest just having whatever shop you select use the paint they normally use for matching factory paint - they're probably going to do the best job when using products/processes that they're most familiar with.