Don't suppose you could share your project (privately?)?..
These are the rules I use; I typically use either a complete testpoint component (so the testpoints are there, physically, during placement and routing), or a reasonably sized via or SMT pad, with reasonable spacing. (The "NoTest" class is there for nets that shouldn't (or can't) be tested, e.g., superfluous single-pin nets, cramped logic pins surrounded by resistor networks, etc. The other "usage" rule you see just allows multiple testpoints in net class "Power".)
The pads/vias themselves are generally assigned manually, since the manager might pick inappropriate or unintended points. (I don't bother with manually assigning fab, I let it do that exclusively.) If all your vias/pads are a unique size, this is very quick with a query.
In this case, test is one side (which is preferred), and pads/vias are only assigned that side. You'll get an error if they're assigned to the other.
I leave all testpoints un-tented, both sides. I doubt Altium checks for soldermask encroaching, at least on the far side... but maybe that's what you're doing that's different and throwing the problem?
Your pad/via size sounds okay, though I might suggest a smaller hole, and optionally a smaller pad, just so you aren't wasting so much space..
Honestly, I don't know how difficult it is to test with large vs. small pads, or what that translates to in test fixture NRE / production costs, so it could be my rules are too tight for a more economically-minded design...
Tim