PowerBook won't start with installed battery

Well you do have a couple of problems here:

First you'll need to replace the PRAM battery if your model has it. Here's the guide to get to it: PRAM Battery & USB Board Replacement. Some of this series had a clock chip which had it's own battery internally on the logic board.

I do think you have a logic board issue with the charging circuit. The DC-In board does not have any of the charging logic on it. This could be a big job to diagnose and repair.

This system is quite old. I think your money & time would be better spent on a newer system. Don't get me wrong here in its day is was one !@#$ of a system! I had the titanium version for quite a few years.