If you want to challenge a takedown notice, the proper (and only) course of action is to file a counter-notice. I had work that I did on Commons taken down by a bogus DMCA takedown notice several years ago. Instead of complaining to the Foundation, which would have been pointless (as they are bound by the DMCA to comply with even the most bogus takedown notices), I mailed them a counter-notice and the work was restored in short order.
There are several handy online guides for how to file DMCA counter-notices. It is very easy and doesn't require hiring a lawyer. The only catch is that by filing the counter-notice you are putting your money where your mouth is and legally asserting that you have the right to post the work (so make sure that this is correct or you may end up in a lawsuit).
The current situation is completely different than the NPG situation, which involved only bogus threats, not a legally binding takedown notice.
Ryan Kaldari