pressure drop makes total sense if the primary isn't running. No doubt the FPCU would typically be bypassed if they're on a staged system like that.
Being intermittent sucks. I don't know, but I'd think once a pump fails, it's completely dead and wouldn't kick on and off all willy nilly - pointing to an electrical break somewhere. Not the most fun to chase, but it looks like you're on track for chasing it down. Good hunting!