How to automate Photoshop's "Auto" Brightness/Contrast functionality on a batch of images?

I agree that Brightness/Contrast is a bit of a blunt instrument. You sbould be able to get more consistent results another way, and also much more easily than what you propose.

Instead of trying to gain programmatic access to the Auto button in Brightness and Contrast, consider creating an action that uses one or more of the Auto correction commands that are already conveniently available on the Image menu: Auto Tone, Auto Contrast, and Auto Color. Descriptions of what all three do are on the Adobe help page Make Quick Tonal Adjustments.

Furthermore, what those commands do is not set in stone. You can adjust the target colors and default clipping values you want, as described on that Adobe page. Some more background about how those work is in an article by Bruce Fraser, Don’t Underestimate Photoshop’s Auto Color. Although written in 2022, the description of Auto Color options still applies today. (What it calls Auto Levels is now Auto Tone.)

If you want the latest and most advanced automatic correction trained using machine learning, then I agree with the suggestion to run the images through the Auto Settings command available in Adobe Camera Raw or Lightroom Classic, if the image file types are compatible. The advantage here is that you can select any number of images, apply Auto Settings, and it will batch process the lot — without having to create an action or write a script.

If you do it with Camera Raw, I don’t mean not Camera Raw as a filter in Photoshop, but Camera Raw as the processor: That is what will let you batch apply immediately. To use Camera Raw that way, you would first use Adobe Bridge to select all the images, then choose File > Open in Camera Raw, then select all the images there. For Lightroom Classic, you would import all the images, then select them all. In both cases you would export corrected copies.

Auto adjustments are hit & miss.  Sometimes they work well and other times they make a mess.  It really depends on individual image exposure and what needs correction.  I see no way to successfully batch this. 

For better control,  I recommend using Levels or Curves Adjustments on each image separately. 

The Brightness/Contrast command is one of the legacy commands that was often inferior to other methods when used with photographic images, compared to levels or curves. Adobe then improved the code and added the legacy checkbox for the old behaviour.

In addition to auto options in the traditional commands, there is also the auto command in the Camera Raw filter, which can be scripted and automated:

var d = new ActionDescriptor();
var r = new ActionReference();
r.putClass(stringIDToTypeID("adjustmentLayer"));
d.putReference(stringIDToTypeID("null"), r);
var d1 = new ActionDescriptor();
var d2 = new ActionDescriptor();
d2.putBoolean(stringIDToTypeID("auto"), true);
d2.putBoolean(stringIDToTypeID("useLegacy"), false);
d1.putObject(stringIDToTypeID("type"), stringIDToTypeID("brightnessEvent"), d2);
d.putObject(stringIDToTypeID("using"), stringIDToTypeID("adjustmentLayer"), d1);
executeAction(stringIDToTypeID("make"), d, DialogModes.NO);

The best solution I can think of is, a .jsx file for Photoshop scripting that opens each image file from the desired folder, adds some default brightness/contrast layer, quite literally sends an arbitrary mouse click to the active window to click the Auto button, waits 5-30 seconds (depending on how safe I want to be to ensure it runs to completion), then finishes the job via more .jsx code to save the image and close the current document, then rinse and repeat until I'm done with the entire folder. Best idea I've got, given that this apparently can't be done EITHER via Photoshop scripting OR via Photoshop actions.