You can use an 11.1v LiPo in full auto, just as long as your internals can handle it.
For example, my G&P M249 is capable of long sustained bursts at 11.1v no problem, because I've set up the internals to handle it.
A 7.4v LiPo, depending on your gear ratio, will give you almost the same RoF as a 9.6v NiMH mini battery.
In addition, because the LiPo offers higher amperage, you're able to get more torque from your motor, and therefore run a higher ratio gearset, increasing your RoF.
So on a stock AEG, install a simple MOSFET switch, an appropriate fuse, and possibly a voltage monitor, and you can abuse your trigger same as you would with a NiMH battery!