Batteries fail in 3 ways
High Resistance (Will have no effect on the converter)
Low capacity (likewise)
Shorted cell (Problem may appear).
Monitor charging current ... Normally if the batteries have been run down a bit you will have fairly high charge current when you first plug in.
But as the batteries fill the charge current drops. Less than 1 amp when full.
A Clamp on Multi-Meter with DC capability (not all can do DC) will indicate the amount of current flowing.. SOME may even do direction but mine does not.