There has been a recent thread about SDR receivers ( unrelated to Fox 1E). I have a yagi/LNA/filter assembly and a feeder to an RSP2 here.
On one of the other forums there was a comment  about maybe listening for the Fox 1E transponder LO or something other than just the beacon. 
Is there a detail block diagram available of the transponder that would call out the LO frequency ? (I'm guessing 581.65 MHz if its a single conversion) or any details of the passband filtering?  Maybe there is enough LO leakage out the front or back end of the transponder to receive.  If there is an AGC/ALC, the TX might be putting out a noise bump that could be seen by the SDR also.

Anyway, just food for thought.
Alan
VE2NHR