The shop is too small? Cause the rack in my picture handles full sized pick up trucks, and my Mini works just fine on it.

I agree with finding a shop that handles high performance or race cars.....

Try a custom shop.  I have noticed a few other shops here in spokane (about 250k) but I suspect they all have the same problem...too small.  The one shop I have used for years (I don't even know how I found him) you have to wait almost 3-4 weeks to get in, and he always wants to take pictures of all the cars. Good luck. 

Just for fun, pic of my 89 Mini on the hunter laser alignment rack yesterday......no issues at all.

In my experience, Mini suspensions stay pretty well aligned once set. The lower portion of the front is triangulated, with one of the connecting points - the long tie rod - bracing in the direction of the major forces. It is in tension under braking, which aids in keeping things aligned. When you hit a bump or pot-hole it just pulls the rod tighter. The lower control arm is almost parallel to cornering forces and works in tension/compression only.
Cars with "A" shaped lower suspension arms have their two anchor points close together and not far from the centre of the wheel. Hitting a bump applies leverage on the two anchor points, pulling the front one outward and pushing inward on the rear one. That's why hitting a pothole can mess up their alignment easily.

i think the very reason why mini cannot maintain a good alignment for a long time is because the lower control arm is just one strait rod, and just supported with another diagonal rod which is then held by 2 round soft rubber imside and harder rubber outside.  IF the lower control arm is designed as "V" shape, like all other conventional cars, alignment wouldve been very stable.  You know kinda like how bridges and church roofs are designed, the topmost middle part is a bit convex, to add to the overall strength of structure.
i wish i was aready alive during 1959 era when minis are being built so i can teach them the proper design., i'll include the drainplug-less radiators in the list.

let's see , when you're at the race track and need an alignment, do we go to the back of the truck and pull out an alignment machine? again, it can all be done accurately with a torpedo level and a good quality string.

I wonder how people align their minis(with 10 inch wheel) back in the days when there were no laser aligning tools yet, do they just eyeball the alignment?

i didnt know that human brain can be aligned too using laser beams... hmmm? i'll get there one day.

Speaking of laser alignments, when I was having radiation treatments, they positioned me to the millimetre using laser beams that came from the ceiling, the walls one each side and from the machine itself. To do so, they had previously tattooed 5 reference marks on my body. Ain't technology wonderful!

Speaking of laser alignments... https://www.youtube.com/watch?v=Bh7bYNAHXxw

Fortunately, we have a laser alignment shop owned by a Mini owner...one who Ice Races a turboed Mini.  No problems with 10", 12" or larger wheels.

I've been doing my own alignments with the following tools for years, and the results are very accurate based on having them double checked by an alignment shop.  

Camber Caster gauge developed by Keith Calver:
//www.minimania.com/TOOL10___Instructions_for_Camber_Castor_Gauge


Longacre toe gauge
https://www.amazon.com/gp/product/B000VAOHB2/ref=oh_aui_search_detailpage?ie=UTF8&psc=1

I used the string method initially to ensure that the front and rear wheels were in alignment relative to each other and that the steering was centered.  The above tools are much easier and more repeatable than the string method for relatively quick adjustments between street and track driving using greased linoleum tiles under the tires.

I have never had any luck using alignment shops. I think the work is to complex and too precision for the average shop employee to get right. Over the years I have built my own simple but accurate alignment equipment to specially fit the cars that I have worked on(Vettes, early Chevys and Minis). When I have checked the work of the alignment shops it has invariably been wrong, sometimes not even close, so I don't use shops anymore. Doing it yourself and getting it right is very time consuming; it helps you get close quicker if you know your Trig formulas and have a Trig-function calculator.....you are not going to do it in an afternoon. I usually spend a few days to a week at it, doing all 4 wheels, then I drive it hard for a few hours to settle the car and check everything again. It is easy to make a mistake. Good luck.

Again, I have no problem aligning 10's, 12's or 13" wheels on classic Minis with our 2 year old Hunter setup with the frickin lasers.....

Modern laser alignment rigs will cope with 10" alloy wheelsbut not steels - you just need to reverse the location pegs to their alternate setting.

13" wheels with their elastic band tyres are a no-no as the sensor plate hits the floor as it rotates.

I checked this on an alignment rig in Dayton two years ago to make sure the same was true in the US.

I donated a set of 12" steelies to my local shop which they fitted a set of 145/70/12s to so they can align any Mini that shows up.

+1 to MiniDave7's last sentence.

I teach automotive courses at the local college, we have the latest and greatest Hunter laser alignment machines. I align my Minis (and my friends) on there all the time - 10" tires and all.

You do have to be a little bit careful getting the car on the rack as the track is just barely wide enough to clear the flanges on the inside of the rack, but it works just fine. With the proper adjustable suspension parts you can really get it dialed in.....

I think the shops that say the car or it's wheels are too small just don't want to mess with it.

Look in the local directory and find a shop that does alignments on high performance sports cars or race cars, they'll know what to do.

BJ at MiniTec did my last alignment using a pair of these: https://www.flyinmiata.com/hub-stands.html
Seems to have worked great, bear in mind that my car has a Mini-Tec front suspension and 13 inch wheels.
But back in the old days, when I still had the stock set up with 10 inch wheels, I always had good luck with the old antique type alignment pit places. You just have to find one where the sign is so old that the Bear on the sign has most of the paint worn off.

6464. Good idea, however, the car has the dump flairs on it which hold the string off the tire on the front and back. Do these have to be removed? I'd imagine without removing it, you'd never get an accurate distance from string to tire. 

Use the string method. Buy a good string, such as a chalk string. Buy a good rule, such as an engineer 6" ruler with  graduations of 1/32.  From there you loosen the jam nuts on the track rod ends. The mini should have been placed on a concrete pad  that is level and straight. put the steering wheel straight. Pulling to the left or right can be caused by worn or uneven tires. Now wrap the string from the tailpipe around the rear wheel and try to get the string crossing the center of the wheel. Go to the front wheels , keeping the string tight and ending up at the tailpipe again. String has to be very taunt. String must cross the center of each wheel. Now  choose a spot where you will consistantly measure the front and back wheel. We are only working with the front wheels at this time. The back of the both front wheels should be 1/32 less than the front part of the wheel. Easy Peasy.