To get back to the original problem, did you check with visible clearance lines which one actually has too large clearance? Here in 5.0RC1+ it’s clearly the via, not pads. Pads obey the given pad clearance.
Can you make the vias smaller?
If the vias are already as small as allowed, you could cheat a bit. Create a footprint with only a round(ish or octagonal) non-pad copper area of the diameter of the needed via diameter. Make the via annular ring smaller and place the footprint on it. The non-pad footprint copper area doesn’t offend DRC. For the power plane part of the via you have to e.g. use a thicker via segment.