I will check with the MFG on the charger. Hopefully I get a decent response and not the CYA response. I know the cell tower generator I once bought had the charger connected all the time but maybe the circuit powering the charger was not backed up (doubtful).
Regarding inside or outside storage, it comes down to how often you think you will use it and if you want to fool with dragging it out and connecting it during an outage. Do you want to be doing that in the sleet, rain, etc.? Are there others in the house who might need to use it who are not capable to connect it?
If it were me and I had a pad, I would put it outside and have it ready to go. Mine is on a military trailer and has been sitting beside the house covered and connected for a long long time.