You are still thinking about the controller and transfer switch the wrong way. The controller and transfer switch are rated for the locked rotor amps of the motor. The transfer switch is rated for not less than 115% of the motor full load current and is suitable for switching the locked rotor amps of the motor. All of the requirements for the controller and transfer switch are in NFPA 20, Chapter 10. Section 10.8.2 covers the fire pump controller and transfer switch arrangements and section 10.8.3 covers the power transfer switch requirements.
The breaker on the generator side is smaller because the generator system is not required to carry the locked rotor amps of the fire pump indefinitely. Regular motor rules apply to that side of the switch.
If it helps in your specific example, think of the ATS as a 2200A ATS that is protected on the generator side at 450A.
I don’t follow you are saying ATS and controller are rated for LRC but then next sentence you are saying ATS is not less than 115% rated full load amps of fire pump? I am still confused. Do you mean utility side ATS controller is rated LRC and on generator not less than 115% of FLA of fire pump? If not then can you please clarify further?