Batch processing

The following script can be recorded as the first or last step before saving in the action. It will create a separate text file in the same path as the processed image, named after the processed image. This way you can see which files have been processed while a large batch is in progress. Set the sort order of the folder by file type or date modified for easy viewing of the results.

/*
Log Filename for Batch.jsx
https://community.adobe.com/t5/photoshop-ecosystem-discussions/batch-processing/td-p/14020439
v1.0 - 21st August 2023, Stephen Marsh
*/

#target photoshop

try {
    var os = $.os.toLowerCase().indexOf("mac") >= 0 ? "mac" : "windows";
    if (os === "mac") {
        logFileLF = "Unix"; // Legacy = "Macintosh"
    } else {
        logFileLF = "Windows";
    }
    var thePath = activeDocument.path.fsName;
    var theName = activeDocument.name.replace(/\.[^\.]+$/, '');
    var dateTime = new Date().toLocaleString();
    var logFile = new File(thePath + "/" + theName + '.txt');
    if (logFile.exists)
        logFile.remove();
    logFile.open("w");
    logFile.encoding = "UTF-8";
    logFile.lineFeed = logFileLF;
    logFile.write(dateTime + "\r" + activeDocument.name + "\r");
    logFile.close();
} catch (e) {
    //alert(e + ': Line ' + e.line);
}

Info here on saving and running scripts:

https://prepression.blogspot.com/2017/11/downloading-and-installing-adobe-scripts.html

Please elaborate. 

But I doubt that Batch itself will allow for a lot of customization, so maybe you need to create a custom Script or plugin.