Well, I assume from the picture it is an M Sport model?
If so, over 100K miles for all those sets of tires? If so, all rubber bushings for the suspension should be checked. Assuming there has been no frame damage or suspension damage, top level Michelin tires, either the best all season or performance, should leave you with a big grin on your face after each time you drive it.
I always recommend an independent tire shop for mounting and alignment, one where the ownwer or top veteran techs do the servicing. Avoid like the plague national or even regional multi store chains that have minimum wage rate pit monkeys work for them. Besides the best equipment, be sure any alignment starts with a 0.0 degree thrust angle. I recommend reading the Tire Rack installer reviews and visit the top 3-5 in your area, talk to them about your problem, and note who does the work.
If this tire shop is "having difficulty with other new BMWs", then that is a big red flag for me. Yours seems to be a 2009 model? I would take it to another independent shop or even the dealerif they are well rated.
Something seems to be wrong with your car, and it is disturbing that no one has been able to identify the problem. Good luck.