Bits to keep: output FIR

If any MSB is not used at all then you need to discard it freely. You can easily know the maximum swing of your output by finding out the maximum input value and the gain of your filter(sum of coefficients in hardware).

If one MSB is used scarcely then you may opt to discard it with saturation.

In principle it is better not to saturate. To avoid saturation you can lift up the filter gain to occupy that bit well or lift down to avoid it completely.

Once you know how many MSBs to chop off then you discard the LSBs accordingly. You may round up for LSBs but this is optional to improve the result.