I have no idea about 4400 geodimeter. Its different platform than these gdm 600 trimble5500/5600.
Loader is active because there is no software in RAM. RAM is made non volatile using batteries on board. Loader is in ROM.
Do you still need the firmware ?
no, i got the latest one already.
It is possible to do a modification adding batteries in parallell when doing the replacement?
Did some googleing. Lithium cells are not supposed to be charged. I guess a resistor could be used to limit the current still having the ability to feed the RAM during the switch to new cells.