What if I were to take a recording, then just extract the frames I needed from it? That should allow for shots much faster than 1 fps. Is it possible that the cameras could be tricked into thinking they were taking a video, thereby bypassing the overhead time between shots?
If the resolution / compression of video is acceptable, then you can use video, no trickery needed. Multicam doesn't have any specific video support, but you can probably start/stop recording on a bunch of cams without too much trouble if you don't care about the individual frames being in sync. OTOH if compressed 1080P is good enough, there are probably better camera options.
You can't "trick" the camera into taking full resolution stills at video frame rate. Video is generally done in special readout modes so nothing after the sensor interface sees the full resolution data, and the rest of the system doens't have anywhere close to the bandwidth required to deal with it.