The only mechanical method of measurement that comes to mind is to use a micrometer, and stick the shaft through the braze-on. This allows you to measure the thickness of the tube on the side opposite of the braze-on. There are a couple of issues with this technique that come to mind... and probably many that don't....

1. the interior of the tube is concave and the micrometer shaft has a flat end. You'd have to stick something convex (like a ball bearing) onto the end of the shaft.
2. the shaft on my 1" mic is a bit over 5mm, and the shaft on my 2" mic is a bit over 6mm. Not going to fit into the braze-on. You could attach a piece of smaller rod to the end of the mic shaft, though. Might be a bit fiddly. The length of the rod could be subtracted from the final measurement to yield the tube thickness.

Just some early morning thoughts...

Steve in Peoria