I modeled up a new mounting bracket based on the remains of the original bracket (in the photo above) and a few assumptions along with plenty of measurements.
The bracket was 3D printed using my Tronxy P802M printer using white PLA with a layer height of 0.3mm and 15% infill. The print took a little over 10 hours. To attach it to the stand I acquired two 1/4-20 by 2-1/2″ bolts with two washers and a nylon lock nut each. To hold the scope in position the bolts need to be slightly tight for a nice friction fit. For the pivot I also acquired a 3/8″ 4″ long bolt with three washers and a nut. This was tighten snugly to provide enough friction to hold the scope in place but still allow it to rotate. I was surprised at the strength of the 3D printed part, it holds the scope which is moderately heavy with no issues. It appears that I was off by 5 degrees with the mounting holes so the bracket does not hold the microscope head perfectly level. I may end up correcting the model and reprinting it but for now I will see if it makes a difference.