In order to obtain reasonable results with amset, the input calculations have to be of a reasonable quality. For example,

The electronic structure (wavefunctions, effective masses and band gap) should be described accurately because each of these are essential parameters that will control the scattering and transport.

The materials parameters (dielectric constants, deformation potentials, etc) must be described accurately because these are used to calculate the scattering rates.

I therefore recommend HSE06 for any calculations that rely on a good description of the bands, such as the wavefunction and deformation potentials.

For dielectric constants this can be more tricky. In general, provided the band gap is large enough (> 0.5 eV) these properties are described reasonably well using PBE. However, if the band gap is small or if PBE gives a metallic or semi metallic band structure when it should be semiconducting, then HSE06 will be needed.

If your PBE results are close to experimental, you could use the PBE results with the bandgap option set to the experimental band gap. I think that is a reasonable approximation to make. This will automatically scissor the bands so that the bandgap is the correct magnitude. Note that I would still run a test HSE06 calculation and see if the effective masses differ against PBE by using the amset eff-mass command (note this requires a uniform band structure calculation on a regular k-point mesh rather than line-mode k-points).