Hi,
It's just the opposite.
A FIR filter works with multiplying and adding.
Imagine 12 bit x 12 bit multiplier, it has a dynamic output range of 24 bits. If you truncate msbs you may cause saturation or reversing the output causing large distortions. If you truncate lsbs you will loose resolution.
FIR filters may cause amplification at some frequencies, therefore you need headroom in the MSBs.
FIR filters may cause attenuation at some frequencies, Therfore - not to loose resolution - you need headroom at the LSBs.
So even 15 bits surely is a compromise.
Trust your compiler and see what happens.
Klaus