I will say 12V and 1,2A is rather 1,2Ah (check Ah mark this is not A).
Ah tell us capacity of battery. Maximum draining current is limited with internal resistance of battery and battery chemistry and battery model possibility. Check manufacturer datasheet for that battery.
7805 (1A), 78L05 (100mA for you Ok) or 78M05 (0,5A) voltage regulator is linear regulator and will make higher losses, maybe look to use switcher regulator. With linear you will waste 0,5W, its too much on longer usage, but if load is used for shorter time intervals and you need simple solution then try linear.
Best regards,
Peter
:wink: