You just need to put a multimeter over it with the engine running, you can replace the battery but it seems kind of pointless as it's most likely perfectly fine.

You want to be seeing 13.xxV with the engine running, I suspect a reading of 12V or lower, if so this confirms no charge!

Depends on if you know anyone decent who can do it for you on an exchange basis or who will recon yours, you could do so yourself if you got the brushes etc to do so.

Martin