11 dB is pretty impressive, especially if you are just looking at the Doppler shift in the carrier.

Which actually begs the question, can you decode anything (e.g. satellite ID) from the downlink/idle signal? If not, how do you figure out which satellite a given signal came from?

My first guess would be that you can fit an orbit to the Doppler shift, find the closest TLE set from space-track.org, and assume that the signal came from that satellite. Then propagate the orbit to the observation epoch and that should give you a fairly accurate guess of the satellite's actual location. Or do you do something completely different?