I have tried BoltzTrap, and EPW but I'm not sure whether Boltzmann Transport Equation is a right approach for a complex system (semi cond. + an Oxide layer or even the semiconductor itself). Also EPW is computationally very expensive.
You can use EPW which will calculate the overall mobility of your system (as far your oxide is an insulator, the mobility belongs to the material).
You are right that mobility calculations are computationally expensive (and sometimes impossible if your unit cell contains too many atoms). But in normal systems, you can do several things to speed up your calculations. For example:
1. Use a coarse grid for SCF/NSCF and Phonon calculations (but be sure that your final Wannier interpolated values are precise enough)
2. Limit EPW calculations to the energy bands close to the conduction band minimum (Valence band maximum).
3. You have to choose fine grids in EPW calculations. However, if you choose very large nkf/nqf it will again result in very slow calculations and massive data. In this case, you can write your own code and implement your own interpolation technique. (For example: https://iopscience.iop.org/article/10.1088/1361-648X/abb2f6/meta )
4. Finally and if you are not interested in writing your own code, there are other codes you can use such as https://perturbo-code.github.io/