Yes that is the way to go, using the SDR as a middle amplifier stage.
Still as you mention there could be some additional noise in the earlier amplification but -130dbm will be a great achievement for sure.
Best to reduce the bandwidth as much as possible should get a much better S/N ration as well.
I was going to consider a PCB filter for the higher frequencies not sure if 1.4Ghz is too low for a strip-line (thats what I called it years ago) filtering system. Maybe that is for another frequency detection.
Frequency shifting for spectral analysis should be easier to obtain at radio frequencies.
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