Calculating amount of base needed to achieve a specific pH in a mixed ethanol-water solvent
I'm looking for some way to calculate how much base I will need to add to an unbuffered solvent consisting of a mixture of ethanol and water in order to basify it to a specific reading (10) on an a pH probe calibrated with an aqueous standard.


I think you have summarized the main issue yourself. The reading of a pH probe in a basic alcoholic solution is just a positive real number. It has no significance whatsoever. The value of 10 means nothing. The pH probe has no idea that it is being dipped in an aqueous organic solution. I am sure the probe will take a very long time to equilibrate to give a stable "real number" output. Since those numbers such as pHe =10 have no theoretical basis it is hard to calculate anything. This is confirmed by a quick look at the standard method.

The ASTM D6423 method says " Readings are taken at exactly 30 s because the meter reading will change throughout the analysis due to glass-electrode voltage drift caused by dehydration of the glass-electrode membrane. Because alcohol dehydrates the glass-electrode membrane, time spent soaking in alcohol solution should be minimized.
4.3 The electrode is soaked in water-based pH 7 buffer between readings to prepare it for the next sample. This rehydrates the glass electrode, a necessary step to preserve the electrode's response characteristics.
4.4 The electrodes are cleaned/rehydrated at least every ten samples by alternately soaking several times in 1 mol/L NaOH solution and 1 mol/L H2SO4 (or 1 mol/L HCl) for about 30 s each. New electrodes are treated by this procedure before first use."

The practical way to "calculate" perhaps is to do a acid base titration. Calculate the titer for each batch with a suitable indicator and add the amount of base accordingly.