Depending on your EL driver box input voltage, you have a few options. All are based on usign a simple P-channel MOSFET as a switch. Note that all values and part numbers are just guidelines, any general purpose NPNs will do, similarly any general P-ch MOSFET will do (as long as it has a low enough threshold voltage, e.g. < 3V or so, and can handle the maximum EL box current easily - check the rating)
Also resistor values are not set in stone, you have considerable leeway with the values here also (the 2.2kΩ R_PULLUP is a guess at the gamepad pullup)
For example, R1 could be anywhere between, say, 10kΩ and 1MΩ (the higher the impedance the more prone to noise, the lower, the higher power consumption and the harder for the transistor in the last circuit to pull down)
EDIT - Okay, now we know the voltages, then we can settle on this circuit. I will leave the rest of the circuits below in case they are of any use to future readers.
This assumes that the EL wire driver supply is always a bit lower than the gamepad supply:
Make sure you choose a MOSFET with a low Vth (turn on threshold voltage) and use a schottky diode (make sure it is the right way round) The EL Driver supply and Gamepad supply ground wires need to be connected. The Switch represents the button, R_Pullup the assumed internal gamepad resistor, and the +3.3V represents the gamepad supply.
Simulation:
Circuits below left for interest to future readers
If the input voltage is +3.3V, then things are very easy, this circuit will work (note - this assumes two 3.3V regulated supplies that are not going to vary by more than a couple of percent):
EL_BOX is a simple representation of the load from the EL driver box. When the button is pushed it pulls the gate of the P-ch MOSFET low and turns it on, supplying power to the EL driver.
Simulation:
If the voltage of the EL wire box is different, let's say 5V, then if you can hack a schottky diode in between the button and it's +3.3V connection, then this circuit would be an option:
The diode prevents the 3.3V rail interfering with the 5V rail, but still allows both to be pulled to (near) ground. A schottky is used as it has a lower drop than a normal silicon diode, to make sure the game controller IC pin sees a logic low.
Simulation:
If it's not possible to hack the diode in (i.e. all you have access to is the +3.3V line and ground line) then you will need a couple of transistors to invert the signal something like this:
You could use N-ch MOSFETs for Q1 and Q2, but bipolars are cheaper so unless you need the lowest power consumption NPNs will do fine.
Simulation:
Something is being missed here. The optotriac is quite capable of switching that load current, however the triac in the optoisolator is handling the inverter's high frequency AC differently from your 'triac'. It's designed to control mains frequency AC - 50 or 60Hz, and is too slow to turn off at the very brief zero crossings when it is fed with ~1kHz AC current
Edit: there is a good app note from Vishay here which makes it a bit more clear. You cannot just use the static dv/dt spec - that is what prevents the opto from turning on when it is off, rather you need the (lower) commutating dv/dt, which may not be given.
In fact, perhaps it's not actually a 'triac' but an alternistor that you're using, which would tend to perform better because the semiconductor structure is different.
The answer in that case is to use a more appropriate switching element, such as a OptoMOS SSR.
Best Answer
The area is not really what loads the driver- what matters is the total capacitance (and the voltage, and frequency). The wires have a spiral wire on the outside which doesn't have as much capacitance per unit area as a flat panel.
The HV816 claims to be able to drive 150nF, which they say is about 42in^2 of EL backlight.
From this link, they claim the capacitance of their EL wire is about 1nF/foot, so 20m should be about 66nF, in which case the Supertex HV816 should be able to handle the capacitive load, and if the voltage and frequency are right, it should be okay. Check the specs on the EL wire you are using (or measure it).