The question title involves a contradiction in terms, because a CIC decimator is a filter with finite impulse response as well.
So more exactly you are asking about using different FIR characteristics than CIC as a decimation filter. The general answer is quite simple, if you want a different frequency characteristic than (sin(x)/x)^n, you won't use a CIC.
CICs have been used and are still used due to there simple, resource saving (no multipliers) topology, but they lack of a steep cut-off characteristic that's often wanted in digital signal acquisition. An economic solution might be a multi-stage design with CIC and a different FIR in the last stage. Or CIC with a correction filter.