I have been investigating the best way to change the battery recently (it arrives Thursday,) and plan on connecting a power supply to the 2 terminals, then desolder the battery, and solder in the new one. Plug the meter in (without sticking my fingers into the socket) and powering it on before removing the supply. Any comments would be appreciated.
Use a fresh batteries like 2 AA instead of mains sourced DC power supply as the backup power during the replacement, its much safer, imo.
Just be extra careful and sometimes shits still happened even for the owner that are fully aware of the risk.
A fine example of an accident in losing the cal at a HP 3468B during the on board battery replacement, which is similar to your 3478A ->
HERE , read the post "thoroughly" why it went wrong and learn.
If you still have any doubt, launch a new thread specially asking the people for suggestion on this particular process, and do NOT rush it just because its holiday, or you're more focusing at the video shooting for bragging purpose, rather than concentrating at the replacement process carefully, its not worth it.