As the battery in the UPS is of much greater capacity than the one in my phone, would the phone (upon connection) eventually be fully recharged, even if the UPS is in battery mode?
Assuming you plug the iPhone charger into a 120 V AC outlet there would be no different than plugging the same device into the wall. 120V 60 Cycle power is 120 V 60 cycle power is the same whether it comes from the wall or the inverter in the UPS. Assuming there is no other device pulling power from the UPS it should
completely charge your iPhone just as if the iPhone were plugged into the wall socket. Assuming of course the UPS battery is fully charged and there is sufficient battery capacity to provide 120 V AC power for up to 8 hours. There is always some power lost to heat when inverting 6, 12, or 24 volt DC to 120 V AC.