Illuminated SPST Switch
I think that you are hooped.  Although it is possible to drive a LED and read a switch at the same time when both are part of a matrix, it's not trivial to do so.  Especially if the LED is intended to operate from 12 Vdc.

Is there any chance that you can open up the switches and bring the LED leads out separately from the switch?  It might even be possible without taking the switch apart - some of the illuminated switches I've seen make the lamp connection right at the bottom of the switch.

You would then drive the LEDs separately from the switch.