What I often do is add a rule like :
IsVia and (Net='GND')
And set the polygon connect to no thermals (direct connect). You have to add a rule at Plane->Power Plane/Polygon connect Style. I think in this you need a polygon rule. Add one rule with the query I added above here on first object. Leave second object at 'All'. Set the connect style to Direct Connect. You would have 2 rules now, one that matches (GND) via's, other that matches 'All' (thus the remaining stuff to connect).
If it doesn't pick it up correctly, you may have editted the wrong rule (the lower priority one of list). You can either swap them, or explicity force the other rule not to work on these gnd via's any more. So instead of the first object being 'All', it will be:
not (IsVia and (Net='GND'))
(I don't think the net check is a must, because via's are normally not soldered or used for components).
This way, there are no thermal reliefs on the GND via's, and so it won't block the internal via's. I can imagine that you want thermal reliefs if you want to solder through these via's to the QFP heatpad, but I think it should be able to be soldered.. I atleast made it work on a MSOP-10 with heatpad that had no thermal reliefs on bottom and top. I removed the soldermask at the bottom and was able to solder it , even with lead free solder. Had to turn the solder iron up to 400C, though.