Should a frequency-controlled motor use a circuit breaker with a B or C trip characteristic?
I assume that you are asking about the branch circuit breaker feeding the frequency controller or variable frequency drive (VFD). There should be no breaker between the VFD and motor. The VFD does not just reduce the inrush current, it actively controls the current and eliminates the inrush. You should check the specifications for the VFD, but VFDs generally limit the intermittent current to no more that 150% of rated motor current. The limit value is generally adjustable to some value below the rated motor current, probably 50% or so. With a VFD, a motor can provide 150% of rated torque for acceleration with the motor drawing current in proportion to torque, 150% current for 150% torque.

At low speed and high torque, the input current to the VFD can be considerably less then the motor current because the motor power factor is quite low and the reactive component of the current is supplied by the DC bus capacitor bank in the VFD. However there can be some high peaks in the input current due to harmonic distortion. The harmonic content is strongly influenced by the source impedance. Higher source impedance (lower short circuit capacity) reduces harmonic content. VFD suppliers often recommend and supply input line reactors to reduce harmonic content. Any add-on reactors must be selected with the knowledge of built-in reactance supplied by the manufacturer.

VFDs generally include provisions to limit power-up inrush current due to capacitor charging.

The first thing to do is check the VFD manufacturer recommendations regarding VFD input fusing and branch circuit breaker selection. I doubt that any will recommend a C characteristic breaker.