This should be a non-issue. Since you are not using the center pad, you should not tin it with solder, nor add solder paste below it. Just do the same thing you would do if it didn't exist.
Use a soldering iron and some de-solder wick to clean the center pad as best you can, and reflow it again.
If it really is a problem, you could try drilling out the center of that dip adaptor. This would give any pooled up solder paste somewhere to go instead of making the chip float.
Rarely will you need or want to put the capacitor exactly on the power pad. Yes, you'll want to keep it close for best decoupling, but just outside the IC courtyard is very good.
As Andy already mentioned, it's generally not recommended to use plated through holes for BGA pads. Even with microvias, there is a tendency for solder to want to wick through the hole and you may end up with voids where your solder balls formerly were.
As you mentioned, filling and capping the via will prevent this, but at added cost.
Most people use dogbones to route out from the BGA pad to avoid putting a via in the pad:
You can make one of these for each pad you need to get to a different layer - typically all PWR and GND pads will connect to their respective planes with one of these vias.
Depending on how big your BGA package is, you may be able to route the rest of the signals directly out on the component layer, avoiding a via entirely.
Speaking of planes, you don't mention how many layers you are planning to use for this PCB. I would say 4 is a minimum for most BGA layouts. If you have a PWR and GND plane you can then use a dogbone and drop a via to connect to the respective plane. Then you can place your capacitor somewhere more comfortable and connect it to the planes with vias.
Eventually you'll need enough layers to bring each signal out without hitting a nearby via (through hole) or trace. Blind vias (vias that go from an outside layer to an inner one without going all the way through) help immensely but again, add cost.
This image from Altium shows a 4-layer fanout for a small BGA:
You can see that you start to have difficulty routing out the further in you go. That page from Altium has some good tips and shows how to use their tools to help.
Mentor Graphics just published a book by Charles Pfeil covering more than you will ever care to know about BGA routing. You can find it here (free download, but requires registration):
https://www.mentor.com/pcb/resources/overview/bga-breakouts-and-routing-5e5d097e-55e4-47f6-b5af-dcc55e1892b0
Altium fanout page:
http://techdocs.altium.com/display/ADOH/Fanout+and+Escape+Routes
Best Answer
For cold or dry solders on BGAs, you are working blind without X-ray or just functional testing like yours, so it's not easy for sure.
First order of business would be to reflow the BGA with hot air. There are plenty of YouTube videos of GPUs being fixed this way in your home oven with some care.
If this does not help and it's your one-off board and not mass production you are trying to solve, I would proceed to step two by putting a weight on the BGA. This will most likely violate IPC but allows for a too small solder ball to bridge a connection (by Z-axis movement).
If this still does not help and you are not interested in reballing the BGA (below), consider bumping the board while hot. Highly inadvisable otherwise, but if you are about to throw away the board anyway, this has been known to bridge the missing connection (by X- and Y-axis movement). I have witnessed a street vendor in Hong Kong who repaired iPhones and resoldered BGAs with this method by tapping it with his tweezer while the solder was flowing and letting the capillary forces re-center it after bridging whatever was not connected. Fascinating!
Last resort, reball.
Lastly, a bit of inspiration for your next BGA project: