I'm new here but may have a little insight.
A lot of Midas and Mieneke's are actually locally owned. They are franchises that individuals buy into. They have the corporate backing for material and training programs for labor. So sometimes it's not a bad deal to check them out. I will say that some shops are corporately owned or part of a larger network, where the franchisee owns several stores throughout a city or region.
In your situation, not too many choices, I wouldn't be too aprehensive to go talk to the Mieneke across the street. Typically they will do custom work, and use the parts you bring in, they just won't warrent those parts or possibly the labor to install. Check 'em out, you might be surprised.
BTW, I'm not affiliated with any shop or anything. I just know this becuase I worked for a Midas many years ago, I dated the owner's daughter and they've remained family friends for quite some time.