You can get solenoids that latch, so they don't need to be energized to hold, then if you're staggering release timings the power requirement is actually really low.
Though I think there may be an easier way. What about something like this: A lower fixed position disk with a pacman slot cut out of it, an upper disk that is motorized and has 15 holes that fit the balloon nozzle and some extra. Position the two disks close enough together that the full nozzle can't fit on end - maybe around half the length of the nozzle. Load it up with balloons by inserting the nozzle at an angle, so with the force of the balloon pulling up, there's enough friction to hold it in place. To release, rotate the top disk and all the balloons left in it, when they rotate over the pacman slot, the nozzle hangs free and the balloon can float up through the hole.
Would probably take some revisions to get right, may have a maximum reliable speed that is lower than you want, and doesn't support launching multiple at once.... but if you can get the physical aspect of it down, the electronics could be a motor with a gearbox, a battery, and a switch.