It is not that simple. It is not clear why you thought your circuit would work when you left out a *custom, four-winding transformer*. Your circuit makes pulsed, low-voltage DC into the wire. That is, current flows in one direction only, in spurts. The output of an AC inverter moves electrons in both directions. This, and a much higher voltage, is required for EL operation. So yes, you need a custom transformer.So the circuit I drew is only missing the transformer which would change the battery voltage to the voltage required for the EL wire, as well as the flyback diode?
Another circuit that is common in small inverters is the Royer oscillator. Here is an explanation:
Here are example schematics: