The charging cct struggling to charge a faulty battery can cause all sorts of issue, better replacing and seeing how it goes, probably OK once swapped tbf.
Most decent installers will monitor false alarms and subsequent root cause, this would then lead to product sourcing.
Most others just buy the cheapest...
As long as it's a good quality detector and there are no temperature fluctuations or draughts then a PIR would be OK, however for the difference I would use a DT.