The 4022's are pre-polorized (electret) condensers, as are AT853's, Nak300's, Earthworks, and many other mics. The power for most electrets is to bias the the mic preamp circuitry, not to polarize the capsule, so it makes sense that it could work. Whenever I hear someone say "electrets suck", I point to DPA's and Earthworks to demonstrate that's not always true.
How well do 4022's perform with 9V? I can't begin to guess. I have some 2 wire AT853's, which are well known to brickwall at a loud show. One time at a typically loud show (Trey) I used a 12V battery box for one set, which gives just enough headroom to avoid brickwalling. For the next set I used a preamp with 8 - 9 volts and it did brickwall a little, it was barely audible, but it showed up clearly when looking at the waveform in Audacity. So in my case that extra voltage helps the headroom enough to matter. Is that the same with the 4022's? I have no idea. Look at the wave form in Audacity or whatever, and see if it looks limited on the peaks. If it does, then 9V isn't quite enough. If it looks fine, than probably it is fine.