Them's the breaks I'm afraid! Putting additional drivers on the policy will help, but you definitely want the policy in your name (rather than being an additional driver on another policy) as you'll benefit from the no claims bonus in the long run.
I know this pertains to an S, but I've put my 2p worth in, as have other people, so you might find some good tips: Cooper S Insurance 19 Years Old
There's a bunch of discussion and example prices here too: How much do you pay in insurance?
Your annual mileage will make a difference, as will your postcode, occupation etc. but at the end of the day, you'll pay a premium for driving a MINI. A 1.0-litre Corsa or similar would cost less to insure (that isn't to say it would be cheap, but it would almost certainly be less), but if you've got your heart set on a MINI, you'll have to stump up some extra for insurance I'm afraid.
Which companies have you tried? Admiral and Bell seem to be fairly competitive on the whole, as do Co-Op and Endsleigh, so definitely try those 4 if you haven't already. I've heard good things about LV for young drivers too. Also, don't rely solely on comparison websites - you'll usually get a better, more tailored quote if you contact an insurer directly. Might be worth doing Pass Plus too - ask a few insurance companies how that would affect your premium.
Good luck and welcome to the forum.