I didn't see any mini specific salvage places anywhere near me (I live in Iowa, nothing is near me haha) but it did let me stumble across salvage auctions in general. I found a place where there are tons of salvage minis for auction that have the same engine and such going for much cheaper than just the engine by itself and seem to be running just fine. I may have to do some more research on how hard it would be to do something like that.
I'm a bit familiar with Adesa which has a number of auto (vehicle, equipment, boats, trailers, etc.) auction facilities. (I wrote automotive test software and had access to the cars at the Adesa site for testing my software.) Some years ago the general manager at the Tracy CA Adesa site told me his facility processed 130,000 vehicles per year and auctioned around 90,000 of them.
Believe there is an Adesa in Des Moines. Adesa is for auto dealers, brokers, not for use by private individuals. One can work with an auto broker I believe to buy or sell in this case a car. Even one declared a salvage vehicle.
Back in 2009 I sold a 2008 Cayman S that I owned but which was hit and declared a total loss after just 4 weeks of ownership. It is a long story how I ended up with the car but it was part of the at fault driver's insurance company's settlement with me. The Adesa facility held a weekly auction of salvage vehicles. The GM put my Porsche Cayman S in the auction and it was sold.
If you think you want to use the Adesa facility to buy a car that you might obtain a suitable donor engine from, or sell your car, you will have to contact the Adesa facility and see how that can be arranged. Or search for auto brokers in the area. A local dealer might be able to direct you to one. Dealers use auto brokers to buy cars at auction or sell cars at auction all the time.