Next question is what kind of battery should it be? Preferably one that is not too picky about getting fully discharged.
What temperature range will the battery be exposed to? Is it OK if the battery has to be replaced after a deep discharge event? How long does the battery have to last at full capacity if no deep discharge event occurs? How soon after power is restored must the battery be ready to function for the full 2 hours again?
It will be from like +30c to -5c.
It would be prefered if the battery could survive to be fully discharged without totally failing, a bit loss of capacity is expected.
The battery is only in case of everything else failing, so it might be connected to power for months or years without having been in use, maybe even its entire lifetime. It is only meant to be used if everything else goes wrong.
Next question is what kind of battery should it be? Preferably one that is not too picky about getting fully discharged. Size is not important, as long as it is within reasonable dimensions.
Sealed lead acid is most common for security applications, but doesn't respond well to deep discharge. Why do you need that though? Are you expecting lots of long power failures? Even if you are, can't you size the battery for 4 hours, and discharge it to 50%?
As I wrote earlier in this thread, it is for when things go wrong. Hopefully it will never be used, because that means something did something unexpected. It got nothing to do about power failures, it got something about to failing to a safe state, but delaying that so it does not happen right away, because that means you could just cut the power and then open the locked door. Discharging to a certain level means there would have to be something to keep track of that too, which also could fail and then there would be a 4 hour delay instead of 2 hours, plus the battery would get too low again.