Hi Juan, a two-battery setup is probably the simplest way to do this. Run the Lilypad off a 3.7V battery, and run your 11.1V battery into the VIN pin on the EL Escudo. Use your soldering iron to put a blob of solder on SJ1; this will send 11.1V to your 12V inverter. Now you just need to connect three of the LilyPad's outputs, to the inputs on the EL Escudo (you can use the row of eight holes near the status LED marked A through H). You should also connect a ground line between the LilyPad and the EL Escudo. Everything should work fine.
Two things I'll mention: when the inverter is on, portions of the EL Escudo will have exposed high voltage, so be careful not to locate it where you or someone else might touch it or brush against it (it's not dangerous, but it is unpleasant).
Also, since the EL Escudo wasn't really designed for fabric projects, instead of going through the trouble of connecting it to a LilyPad, you might consider just plugging a standard Arduino (such as a 5V Pro) into the EL Escudo. You could use one battery (your 11.1V) to power both the Arduino and the inverter. It will be a little bulkier, but because they just plug into each other it would be quite a bit easier to set up.
I hope this helps, good luck with your project!