Yes, you can install the FIPS firmware on a non-FIPS controller. Please 'write erase all' after you did that. The difference in controller hardware between FIPS/non-FIPS is physical to prevent things like accessing the chips via the vent openings, tamper evidence labels (TEL) that needs to be applied, and some supply-chain differences.
For certified FIPS operations you might need those features as well if you only need to run FIPS software that is possible with the same software features that are required in FIPS operations. Without the FIPS controller hardware, you probably can't claim full FIPS compliance.