No, the keg does not have to be full to allow for carbonation.
I've never messed with a sanke keg, but from what I have heard/seen, they can be a pain to refill due to a PITA snap-ring and such. But if you have already figured it out, then ignore what I just said.
As for pouring all the bottles into the keg, I would highly recommend against that, due to the HUGE risk of oxidizing the beer, and thus making it undrinkable. Maybe you should save the keg for your next batch of beer?
You could try using carbonation tabs in the beer with low carbonation. Just carefully open them all back up, add the amount of carbonation tabs that would bring the carbonation level to where you want it, and seal them all back up with new caps. I have never used carb tabs, so I'm not sure what the amount-per-bottle is, so research should help. Or maybe you could even add a very small amount of cornsugar to each bottle, but this may give you some very uneven results.