Can the overlap-add method be used with varying cut-off frequency filter?
The problem here is not the application of the FFT for filtering but the fact that you want to model a linear time-variant system using methods for linear time-invariant (LTI) systems.

If you used the linear convolution instead of the circular convolution + overlap-add you would still face the problem that you have to divide the audio signal into blocks of data during which the impulse response is constant.

One solution is to make the data blocks sufficiently short so that the impulse response can be considered quasi-invariant during one data block. For example, if your audio sample rate is 44.100 Hz and your FFT length is 1024 every impulse response change needs to be approximated by 43 discrete steps per second.