So I was wondering why Mode A and C transponders are not supported?
Given that you have an on-board GPS unit, and you know the RSSI from other Mode A and C transponders, and given that the aircraft is moving through the air, localization should be possible even for Mode A and C transponders by using a simple free-space path loss propagation model (the air doesn't have many reflectors anyway).
Slides 10 and 11 present a simple solution:
http://www.eit.lth.se/fileadmin/eit/courses/etin10/2014/LectureA3_Positioning.pdfE.g. the RSSI for the last X known positions could be used to estimate a static position, and since the other airplanes are moving through the air the next position could be estimated by adding the newest sample and discarding the oldest one (rolling estimation). Kalman filtering could also be applied to get rid of most of the errors.
/Dimitrios