Reduce output noise power while keeping input bandwidth constant
Is there any way to somehow decouple the input bandwidth of the system
  and the noise power fed into the system, so that the power of the
  noise would be independent of the input bandwidth of the system seen
  by the source?


Yes, it's called filtering and specifically low pass filtering. However, if you want to measure valid signal frequencies above the filter cut-off you have a problem - noise is filtered just the same way as signals.

On the other hand, if you know your signal frequency then you can build a filter that tightly hugs that frequency and, to a reasonably large extent, removes noise except that which is in the pass-band of that filter.

This is how a radio receiver operates - it filters the input to just the approximate bandwidth of the modulated signal and all other noise is fairly well excluded. It's most noticeable on radio receivers that quote a sensitivity figure (usually around -90 dBm). If the radio has a tighter bandwidth, that figure can easily be in excess of -110 dBm because the filtering can be much tighter.

Of course, if you can freeze the signal source (and the noise previously produced was thermal in nature) then this can help a lot but you might not be keen on a bunch of helium or nitrogen tanks lying around.