Does it make sense to 'deflate' a sectoral price index in a regression analysis?
In that situation, how do you extract these fluctuations related to
  the general economy out of your price index being used as an
  instrument?


Regress the sectorial price index on the general price index, and use as instrument in the demand equation the "residuals plus estimated constant" from the previous regression.  

These residuals will be by construction orthogonal to the general price index which indeed could be also a demand shifter, something that would invalidate the sectorial price index as an instrument.

As is the case with any time-series data, one should also deal with possible issues of non-stationarity, co-integration etc.