Think I've got to the bottom of my problem. The EW-SW100 satellite on/off switch on one of the bikes is bad. I've cross-rigged the switches from each bike to the other and the problem moves with the switch.
@puma931 not sure if this is anything like your problem. My symptom is that the bad switch still flashes green when pressed but doesn't switch the bike on. Nothing to do with the battery. The batteries were all fine having charged to having the single green led solidly on.
The wiring for the bike that didn't work had a coils of excess cable shoved into the base of seat tube without any cable tie. The wiring for the bike that worked looked like a lot neater like this...