In the UK 'Bat' can also be short for 'Battle axe' - a formidable mature lady, who rules the roost.
(if you have ever seen 'Last of the Summer Wine' from the BBC, (excerpts available on youtube) most of the ladies there would qualify.
I personally wouldn't claim I was 'mature' enough, but I expect the primary kids at school probably think so.
Rosemary (trying to be an immature SoTTt and Bat in Britain)