I haven't tried any of this, but it is an idea that I think may resolve your problem.
1st Define your test point pad as an oval shape with an offset origin (similar to sketch included)
2nd Then on your design if you place this oval test pad onto a via, it will locate at the cell origin.
Via will be attached to the test pad with no track and can be probed onto left hand side of test pad - so not as to probe direct onto a via.
You can allow vias under the pad in the editor control but you should create a unique pad in the padstackeditor for testpoints just in case same pad is used on SMD components to avoid accidental vias in them. (can also place a via obstruct in center on the cell)
Though for some reason it does not allow to place the testpoint with an offset on the vias but it does work if you place the testpad next to the via, route a trace and then move the testpoint onto the via... (or place with drc off)
Working with offset pads seems a bit dicy because the testpoint center to center will not be chekced properly when using different rotations and not sure if all downstream tooling will use pad center and not testpoint part coordinates.
Thanks, I appreciate both responses. Both of you have been more than helpful.