Here are some results. I can plot other stuff if requested.
Keep in mind that this is only a simulation. Component tolerances, model inaccuracies, etc., will probably result in different absolute numbers, but the overall operation is pretty clear to me now.
I originally tried simulating by using a very large capacitor (1 to 10 Farads) as the battery. This worked pretty well, but I then changed to using a PWL voltage source as the battery, sweeping it from 12 to 15V.
I took a few liberties with the schematic, just to simplify it. if you see any problems, let me know.
Note that the response gets a little mushy with the pot at 100%. I think this is because the supply voltage is not quite high enough for this setting, but it shouldn't matter, as the set point is probably too high to be useful.
These results change if the supply voltage changes, which sounds like a poor design, but the simple car battery chargers rely on line voltage being predictable and constant, so maybe this is OK.