I bought one mini sight unseen from someone that was recommended by numerous members of this forum and a respected member, after asking basically the same questions as you. It was a huge mistake. The pictures looked great, the car was a rusty piece of junk. A partial list of the problems the car had, worn out syncros, brakes that were down right dangerous, tires that wouldn't pass inspection, electrics that were so bodged it would be laughable if it wasn't so dangerous, and then the rust.....

To also quote Bill Cox "never buy any car without seeing it first" He's right, because even if you find someone to inspect it for you their idea of "a no big deal repair" and your idea of "a no big deal repair" can be worlds apart.

My current mini I drove to see it, drove the car, jacked it up, and went over everything about the car with the owner and nearly passed on it, I bought it because the problems it had I knew were easy to fix and the owner reduced the price. This was after I had looked at probably 6 different minis.

I've flown and and looked at cars and not bought them. Was very glad to spend the money on airfare and hotel - saved me a lot!
I've agreed price based on descriptions, flown out, inspected and purchased.
I've only ever bought one that I didn't put eyes on myself - I asked another member of the Mini community to inspect it for me.

Probably the most costly and time consuming thing on a Mini is body/rust repair. If you can't tackle that yourself, it's going to cost a pretty penny so I would pay EXTRA special care in that area. Pix with the carpets removed, from under the car, around the subframe, scuttle panel, boot floor, battery box...bascially all of it. I bought 3 car sight unseen. 

I have bought cars sight unseen and it's a gamble. Bought a cheaper TR6 off eBay about 15+ years ago and it is a total pig. So much bondo and coverup. For the price I wasn't expecting too much but the seller totally hid a lot of issues. Be careful.

I bought sight unseen from a member on the forum, PoloGrounds, and have no regrets. You could request a start up/drive video from the seller, similar to what BAT does. 

A forum member Hugh Cannon AKA Dr. Mini has had a long time occupation travelling the US moving Minis, parts etc from one place to another. He might be a suitable service to transport a Mini from the seller to you. I'm not sure but he might be able to cast an opinion if he happens to be in the vicinity of the seller.

Bill Cox, an experienced and knowledgeable importer and dealer in classic Minis and other vehicles *and former member of this forum) says"

"re buying classic mini sight unseen in Texas I have used www.lemonsquad.com with good success.. later bc"

You would hard pressed to find a shop to look at a car. Minis have not been sold new in the US since 1967 so any shop that might be close would be like yep, we work A mini once a year maybe. Better chance that someone one here or another mini forum would be a better bet. If the ads you have been looking at are classic cars dealers then they probably know next to nothing about the random mini they have. I bought my mini sight unseen from New Zealand. But I had someone I know well vouch for the seller.

I live in north Texas and am talking to several Mini sellers here who live elsewhere and through ads on Hemmings and Auto Trader.  I've bought motorcycles that way; one perfect experience, and one that ran half a mile before I walked it back to the seller and Uber'ed my way back to Amtrak to go home.

That said, have you bought sight unseen without going in to see it?  Did you just trust the seller or pay a local shop to bless the car.  If you used a shop, how did you find a trust-worthly shop?