I am not sure why you'd want to fill the vias.
However, maybe it's worth a try. Everyone always insists that they will wick solder away through capillary action. However, on the internet, everyone will generally just repeat what one person stated somewhere, often wrong. Pretty much everyone says that you can't do BGA at home, you need an X-ray machine, you need solderpaste, etc.
First of all, make sure it's worth the trouble. You can only pass one wire between the pads anyway using 6mil wires - anything smaller will cost a lot. So if you drop too many connections to the bottom with filled vias, you won't be able to route them out anyway. You may find that it buys you nothing.
So let's say you want to fill the vias. First, you'd have to make sure there is no soldermask. You'd have to do add a 'filling' step prior to placing the BGA. To apply an even amount, you can either float the board on a layer of molten solder and let the wicking action pull solder into the vias. Alternatively, you can apply solderpaste and let it get sucked in.
I'd go with solderpaste - it seems more precise. Maybe from the bottom. You will need a lasercut mask with really small holes. Then you will have to make sure that the surface is even - if some balls don't touch the surface because they are raised, they may not make proper contact.
Once again, I certainly would not do this - I've lost enough 700ANs already
Pretty soon I'll start a 'reballing' thread...