I have never seen a simple closed form solution to that problem, without going to the EM simulator. I agree, HFSS stinks and is WAY overpriced. I haven't used it in years, but remember a tendency to not converge on these types of problems, waveguide is big, probe is small, so you need a very fine grid to get correct simulations, etc. The main problem in forming the closed form solution is that you are impedance matching between a TEM coaxial 50 ohm impedance (V/I) to a TE mode waveguide impedance (E/H), and the EM simulators handle this sort of problem in stride.
You could probably arrive at a good design empirically, especially if the frequency is not too high. I would get a good waveguide load and hook it and a waveguide straight section to your transition. Then I would hook the coax line to a network analyzer and watch the coax port's impedance. Sweep the frequency to see where it seems to resonate (where the backshort is the right distance from the probe). Then at the center of resonance, see if it is 50 ohms. If it looks higher than 50 ohms, I would make the probe a bigger diameter. If it looks lower than 50 ohms, I would make the probe a smaller diameter. After you exhaust the diameters in your tool box, try making the probe a little longer or shorter. When you are done, if the center frequency is off, move the backshort around, closer for higher frequency, farther away for lower frequency. When it looks like 50 ohms and on-frequency, measure whatever dimensions you have!