a 1990 E34 cluster does not have batteries, that was on older cars like E30, E28 etc.
PPPP means there is no communication between the cluster chip and the OBC. Remove the cluster and check the pins at the back of the cluster.
What can happen is that the capacitors get old, and these can be replaced in the cluster, see here
BMW E32 - Dash Capacitor Replacement
Intermittent power window and wiper faulure can be caused by a haircrack in the fusible link
and apparently your mechanics found it
BMW E34 Website
Also check the speed sensor on the final drive
BMW E34 Website
As for PPPP: solder in new capacitors in the cluster and see if that cures the problem, if not, maybe you ask in a forum specialized on E34 like
BimmerNut.com Forums
or bimmerforums or bimmerboard E34 section
As for the battery:
Max closed current for Z3 is 30mA