That's what we did with our December 2015 F54 Mini Graham after the 3 year free servicing contract ran out. The problem is its unlikely independant specialists will have BMW head office backing so will be unable to update onboard service records despite having the OBDII tablet that allows updates on coded parts, resetting charge profile after new battery etc. This leaves you with a frustrating service overdue warning at every start up.
Should not be the case. I started to use a local specialist once the good ladies R56 got to 7 years. A small but keen bunch who knew them very well near Basingrad. They had registered with BMW/Mini and had access to do the service updates and register any changes. It was a cost on the service (£20~ from poor memory) but compared to the saving not really material to the overall cost difference. It is the same on my Merc S212 with a local MB specialist who also have access and update the factory records.
I think there could have been a fight between the auto-trade and vendors for restrictive practices that made it possible, under the correct conditions, some time ago.
I know both parties have to sign up on a contract with BMW/Mini and MB to have this access so if your local garage does multiple types it may not be worth it for them.