Usually by the time you need a compressor, you need a receiver/dryer at the time because they get contaminated by debris from the compressor. A new compressor is the right idea, I don't like rebuilds unless I do them myself. I don't have an estimator up but I can give you an estimator link:
https://repairpal.com/estimator/mini/cooper
Here's a directory link for a shop:
https://repairpal.com/mini-repair-in-glendale-arizona
RepairPal Certified shops can diagnose and repair your 2004 Mini Cooper. They provide transparent pricing with a minimum 12-month or 12,000-mile warranty on repairs and services.