I'm on mobile at work, I've confirmed with an EE here that does this sort of thing all the time.
The good news is that your easiest bet is to just ask the board house to fill the vias with epoxy, they can plate over that, and it's very common these days. He said that filling a normal via (not blind or buried) with epoxy is going to be cheaper than what I previously proposed (using blind vias).
The better news is that you can actually rework your current boards by filling the offending vias with epoxy, if they're otherwise usable. A pneumatic shot dispense system would be needed but they're cheap and available now thanks to China.
There isn't really any bad news. He said it's extremely common, we do it probably 100's of times on our boards at work , which are incredibly dense and expensive. No issues at all, it's extremely reliable to do this, even across temperature ranges and vibration.