Sub Band DFT response
Generally if you want to use the same number of bins for a sub-band you should increase the length of the FFT accordingly. That is, say you sub-band width is $BW \;Hz$, then you should increase the FFT length by a factor of $2048/BW$. However, as you can see, this only works if $2048/BW$ is a power of $2$, for example in the first case you mentioned ($768-1260 Hz$). Otherwise, it would required resampling of the signal.