Your highest frequency of interest is 11360Hz, so a sampling rate of ~30kHz is enough, given a brick-wall
analog anti-aliasing filter is employed to prevent frequencies above ~15kHz folding back.
One approach is simply to go to 40kHz sampling rate anyway, avoid the anti-aliasing filter, and then
run several bandpass digital filters on the samples to pull out the channels and separately calculate
running rms power estimates for each band. With crude low-pole-count IIR filters for each band this
might be less computation than FFT or FCT
However for true A-weighting you kind of need the detail of an FFT output produces to do the weighting properly
if accuracy is important, since the weighting varies appreciably across each octave.