The MINI W11 engine uses an Eaton supercharger. MINI does not recommend oil services, therefore no oil is available through MINI parts. However, General Motors uses a similar Eaton supercharger and their oil can be substituted. GM Part number 12345982.
The Bentley Manual shows you home wrenchers how to check and top off the supercharger oil on 2004-2006 models in
this article.