How about trying to introduce phase sensitive signal detection ( lockin
amplifier). I guess the idea is that you wiggle the feed so that any
signal is modulated by the the wigle, but the sky and thermal noise is
not modulated. The wiggle would have to be wider than the antenna beam
width. You could do your FFT then incorporate a digital lockin for each
frequency bin with the known wiggle frequency. It is unclear if this
would realy give you better SNR than just extended intigration, or more
points in the FFT.