Valid approach to implementing a band pass filter
then you'll be doing either "Overlap-add" (OLA) or "Overlap-save" (OLS) which is sometimes more clearly called "Overlap-scrap".  you need to read up about this.  

then simply "apply[ing] gain in those bins" is multiplication in the frequency domain, which means convolution in the time domain.  if your frequency domain function is a rectangle function (some number for "those bins" and another number, like 1, for the other bins).  then you need to compute the impulse response that goes with it and make sure that FIR impulse response is not too long for your OLA or OLS.  do you have a good book that describes OLA or OLS?

but a bandpass filter in the time-domain (your option 1.) is simple and can be implemented simply.  maybe more difficult to design your IIR or FIR bandpass filter.