Should I bother? HP-34C in rough shape

[AndiGer]

Never thought about it is dangerous. As long as you don't eat it and wash your hands after working on it.
The broken cathode after moving the flex cable is often and the Achilles heel of the spice series. The flex will break at the border where the strips are coated.

I think you should find someone - a friend or neighbor maybe? - who can help you with the soldering. The calculator IMHO will work after this fix done.
As long as you can charge the battery outside the calculator as already mentioned no need to re-connect the river-left most connection - because there is no obvious place you can solder to without further disassembly. Otherwise you must remove the calculator's guts to remove the upper circuit board. Then you have the chance to clean the keyboard :-). Then be sure not to loose the 2 switch contacts.
Good luck