So can we assume that your garage is not near your house therefore that's why you need the 12 V supply.
I would suggest at least some form of solar charging and possibly dual batteries.
A few years ago, we needed to run an electric fence for horses to stop them eating too much grass, and I used three leisure batteries, with one always on charge at home and the other two powering the fence.
I also fitted a device which cut off the supply when the battery voltage drop to below 10 V as this prevented the electric fence from draining the battery supply and completely buggering the batteries.
Like anything your alarm system needs to be reliable and to give as few false alarms as possible, or you will end up ignoring the real activation.
Would it be possible to use some form of wireless network with a GSM Sim card which had data on it which would also give you basic camera facilities so you can see if someone has broken into your garage, and possibly even send images to a Google Drive.
I use Samsung smartcams, but also have some software called I spy which has both a free and paid versions which record information from wireless cameras and then send it to the Internet, However all these facilities and features will require more power.
The reality is it doesn't matter, what you put in place it is possible to circumvent it, if ATMs can be ripped out the wall with JCBs or 4x4's most garage doors will offer little resistance.
However your alarm system will at least give you some peace of mind and if possible some extra security in terms of strengthening doors and windows, will also be money well spent
Sent from my iPad using Tapatalk