I'm wondering how much inaccuracy will result when I can't enter the short standard L1-L4?
I wrote a small Octave script that calculates the S11 response of open and short cal standards for a given set of parameters (in Keysight format):
https://www.mariohellmich.de/lostfound/files/ck-model. You have to modify the script (enter your parameters, frequency range, number of points) and run it. The script will generate two Touchstone (.s1p) files for the open and short. Then you can plot the S11 response of the shot, both with the specified parameters, and with the offset inductances set to zero. You can use an external Touchstone plotting software, or modify the script to use Octave's own plotting functions. Check the phase difference of both cases. I'd bet they will differ by at least a couple of degrees at 26.5 GHz when the short is not fully corrected.
Another question is what the impact on the measured S-matrix of a DUT would be, but I think it is safe to assume that phase accuracy would be uncertain by at least be the same amount.
Or, would it be possible to correct for the difference in post-processing on the PC-side, in case more precision is ever needed?.
Not exactly by post-processing, but you would have to do the calibration offline. That is, measure the cal standards with your test setup (open, short, load at both ports, and a thru for a standard SOLT), and then the DUT. Then export the data to a PC (that makes eight Touchstone files), and do the math as well as the plotting. Of course, that could be automated via GPIB.