I don't think we used to take that many spares when we did the moke trips thru the out back
How often is your mini serviced? How old is the motor (as in when was it last rebuilt.... if ever?).... does it tend to give you trouble? How often do you take it out, and what kind of driving do you do in it?
If its serviced at normal intervals, and you check everything over each time, and its not a naughty mini most of the time, you should be fine

just take a 7/16" spanner, a 1/2", a 9/16, a Phillips screwdriver, a flat blade screw driver and a pair pliers.... and """IF"""" anything goes wrong you'll have the tools you need.... but if its serviced normally you shouldn't need to take a second support vehicle just to carry parts

besides, you're on main roads for most of that trip, so you'll never be far from help
Hope you have a blast and I'll probably see you on the road
<edit> check your wheel bearings before you leave

thats not fun! and yeah, to be fare I always carry spare fuses in all my cars, and tools and a torch... Just give your car a good inspection before you leave....
Tyre pressure
Wheel bearings
brakes (fuild and pads/shoes)
look at your CV's
change the oil + filter
check coolant
check air filter
and you should be sweet
