How can I do a frequency sweep when the frequency modulation index is dependent on the amplitude?
If we go back to basic theory, we have a carrier signal of the form :-

\$E_c\cos\phi_c\$

... and a sinusoidal modulation signal of the form ...

\$E_m\cos(\omega_mt)\$

and if we let the frequency deviation be proportional to the modulation amplitude, so

\$\Delta\omega\propto E_m\$

the instantaneous frequency is given by ->

\$\dot{\phi_c}=\omega_c+\Delta\omega.\cos(\omega_mt)\$

Integrating this to get the instantaneous phase ->

\$\phi_c=\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\$

So the modulated output is ->

\$E_c\cos\Big[\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\Big]\$

As you say, the modulation index is dependent upon \$\omega_m\$ so the relative amplitudes of the spectral components will vary with \$\omega_m\$, but the modulation index is also a measure of the peak phase deviation, so if you want the spectral amplitudes to be independent of \$\omega_m\$ you must have \$\omega_m\propto \Delta_\omega \propto E_m\$, ie phase modulation.

One technique of producing phase modulation is to use a frequency modulator with pre-emphasis of the modulating signal to get the amplitude proportional to the frequency.