Since the Transverse Mercator projection is essentially a rotation of the standard Mercator projection cylinder I suppose one hacky option is to introduce that equivalent rotation into your rotation file. Perhaps as a new plate ID relative to the normal anchor plate 0. And then select that new plate ID as the anchor plate.
Then use the Mercator projection in GPlates. However the grid lines would look wrong, so you’d need to turn them off (by setting the alpha channel of the grid line colour to zero). You could then create the grid lines as regular geometries yourself and load them in (they would also be correctly rotated by the non-zero anchor plate ID assuming they are assigned plate ID zero). I think someone has done this before but I can’t find it now.
In GPlates we’ve stuck with pseudo-cylindrical projections since they have global coverage, and based on a vertical cylinder since we can then do dateline clipping (around the longitude/latitude rectangle) so that things look correct in 2D map views (including exporting to other software like ArcGIS). Also, in the next GPlates supporting Vulkan graphics, the map projection will be hardware-accelerated (first converting 3D position to 2D lon/lat and then using 2nd order interpolation via textures into map projection space) - that means it’s fairly invested in that class of projections. So we don’t really support other projections (like you mentioned in your other thread).