The basic fact is it will use 7.5 watts to energize the coil, then 2.5 watts to hold the coil energized, as all you are going to be doing is energize the coil and then release it, only the 7.5 watts if of interest here.
7.5 / 12 = 0.625 amps.
Due to the fact this will operated for less than a second on average, to would add upto not a lot of amp hours, but compact size for 12 volts is the problem.
It is likely that it will work off of 9 volts but the amps go up (7.5 / 9 = 0.833 amps) and you might get 100 button presses from a battery.
You really need to find a solenoid the is rated for a lower voltage, also remember that most solenoids are made to pull in and not to push out as you require.
Pete.