That pretty much sums it up: got a woodscrew through a tire that is only a couple of months old. It leaks slowly. Are these possible to repair? I dread what the replacement cost might be. Over 20 years of driving and this is the first tire I’ve had to replace from “road damage” (versus wear and tear).
From quickly scanning the net, it appears that a tire shop might be able to repair this, but only if it’s not been driven while uninflated (I’m assuming that this is not a self-sealing run-flat, but one that is self-supporting --> link). The biggest difference in a run-flat tire is the very stiff sidewall that will support the rated load, at reduced speed and for short distances, after loss of inflation pressure. You might try calling local tire shops, and explaining your circumstances.