FFT for a specific frequency range.
I think the best solution to your problem is to use the chirp-DFT. It's like a magnifying glass for a certain frequency range. It is more efficient than the direct implementation of the DFT (without FFT), because an FFT algorithm can be used with some appropriate pre- and post-processing. You basically need to modulate your signal with a chirp signal, then filter using an FFT, and then again chirp-modulate your signal to get the desired frequency response. See here and here for details on how to implement the chirp-DFT.