I seem to recall that the controller uses temperature to work out when the batteries are fully charged, hence the instructions on the temperature range for charging and why not to put the controller on anything that generates heat when charging.
I'm at a bit of a loss on the description of a power supply and then fitting replacement batteries. Did you only have this supply attached while you fitted the new batteries? I can't think of a reason to do this. I'm pretty sure the HC1 and 2 don't have anything stored in volatile memory except for what you might be immediately doing, such as turning the throttle, blowing the horn, Everything configuration wise should be saved in non volatile memory. Pictures of what you've done would definitely help.
As to the batteries, I imagine the brand and capacity is largely up to what Ring has on hand on any particular day, the batteries in my HC2 are Rayovac 2500mA. Better quality batteries which the Panasonic Eneloop might be (only seen 2500mA, not 2550mA) probably won't give you any more run time as such but should help make them last for more recharges.
- Tim