Hey,
I work for the company, etrailer.com, that sells bike racks and hitches in the St. Louis area, out in Wentzville. We carry the Yakima SuperJoe 2 bike rack and the Curt hitch that may fit your vehicle.
If you like, you could come to our store and test fit the SuperJoe 2 bike rack to your Mini to see if you like how it fits before buying it. Also, you could bring the hitch bike rack you already have and we could see if there is a way to make it work with a hitch.
Regarding the recommendation for using straps with a hitch bike rack, I can call Curt, the manufacturer of the hitch that fits your Mini, and find out what their recommendation is for using the strap on your vehicle. I would need your Mini's model and year to call the manufacturer. We did an installation of the Curt hitch on one of our other employee's 2007 Mini Cooper S not long ago. You can see the video of installation and the final product on our site, too.
Also, I drive a Mini! They are the best cars! I love her!
Let me know if I can help! My email is kristin.bross@etrailer.com.
Kristin