Hehe, glad you spotted the problem.
Sometimes DC into AC it can be a mistake and damage something, and sometimes it just won't enable the device to work properly, but in this case it's no problem at all.
The designer may even have made it like this on purpose to avoid problems with people using wrong polarity adapters, and also to enable trouble-free 12V use.
I see the battery holder is a loose item, and there seems to be enough room for you to unscrew the holder and put the 12V battery in there instead.
Then you'll have to connect it to the red (+) & black (-) battery holder wires somehow, and also fasten & insulate it so it doesn't move around and short out.