Buying a float charger would be the best way to ensure you don't repeatedly damage batteries, but it could be the case that 1 month of staying unused might be not too long, assuming when you do drive the car, it's a good long drive (or several drives) and not something like 30 minutes and then 1 month of parking again.
Lead-acid batteries do self-discharge, and cars also have parasitic loads. What the parasitic load is, varies from car to car. For example, my car (Toyota RAV4 hybrid) has a keyless entry system. The keyless entry system all the time transmits ping signals to the key, to get a response. The problem is, those constant pings slowly deplete the battery.
Thus, there is a battery saving scheme:
- If the smart key system hasn't been used for 5 days or longer, its response time becomes very slow. So it will take a lot of time to unlock the doors, and it's not as instant as it usually is.
- If the smart key system hasn't been used in 14 days or longer, only driver's door works for unlocking and not the passenger's door
If your car has a smart key system without such a battery saving scheme, it's plausible that in one month, the 12V battery could be flat, or if not completely flat so deeply discharged that the deep discharge might have shortened its lifetime.
As for battery specs, there really aren't much to choose. Your car most likely has a certain size battery holder, and that limits what capacity you can buy. Bigger capacity would of course handle longer parasitic draws, but that's impossible given the limited space. So the only choice you may be able to do is by different types of battery (flooded vs AGM); AGM doesn't dislike deep discharge as much as flooded, but it still isn't a good idea to deeply discharge AGM.
