When you've got something like this it can be beneficial to flux the surface of the board as well as the solder braid before heating it. You may need to put a bit of pressure on it to ensure it makes proper contact with the via. Even then it can be difficult to get the solder out depending on your flux, iron, braid, etc.
If you can't get the solder out, your best bet is gonna be just to add a bit of solder to the vias and solder to them like they're tiny pads. This leaves results that aren't as solid as the through-hole method, but should be fine so long as you don't have any cold joints. This would be less risky too since you won't risk damaging the via.