You can do it yourself. Get a good string. I use  replacment snap line.  get the mini on a flat surface. Center the steering wheel. crawl under and loosen the  stop nut of each side at the tie rod end. don't jack the mini up to do this. wrap the string around the entire mini connecting all four tires, pull tight. Try to get the string at the wheel centers. Now measure the back of the front wheel for a space of 1/32. You will need to adjust the tie rod to get the desired measurment. By getting 1/32 on each side will get you the 1/16 toe out the book calls for. Don't forget to tighten the stop nuts when done.

Find a shop that does race cars.

Just drag around a spare wheel/tire, a tape measure, and the dimensions of your car. (wheelbase, track, minimum width between the ramps on the alignment machine, etc.)
It sometimes helps to have the conversation go something like;
"I want to get my bland boring jelly bean car four wheel aligned. What do you guys charge?"
Insert $ $ $ amount here.
"Well, that isn't too bad.... You guys ever do any old British cars?..... What kind? Well I happen to have the dimensions right here and a spare wheel to make sure you can hook your equipment up... Let's go take a look at your alignment rack..."
When you're in the back with the tech, sometimes it's helpful to ask what his favorite flavor of beer is....

Or, you could find a nice flat spot, a trailer comes to mind, and set up strings... Lot's of ideas out there if you google "string alignment"

I hate ask another question like this but I am really about to loose it. I now have my Mini ready for the road, just waiting to get it all setup and I am stuck. I have been dragging the car from shop to shop with no luck. I give these guys all the figures/dimensions, they say they can "no problem" but I inevitably get the call saying its too small, or the wheels are too small...I think I have spent more on gas towing it around then it would cost for the actual alignment!!!

I am looking for someone in NJ (Central would be outstanding) that can handle this. Need all 4 wheels done (camber/toe/caster in front, camber/toe in rear). Does anyone know what kind of questions or equipment I should be looking for. Car is on 12" wheels. Thanks so much! Sorry for a repeated topic.