Auto levels - White point only with action/script

It is not clear what you want to say and what your screenshot means.

Auto levels uses some computer algorithm I know not what. To Adjust the white point you use white eye dropper  or set the white point down some from the default 255.  That you can save as a preset. That would be a custom setting not an auto setting.  So with that in mind  I set a level adjustment manually where I set the black point to 20 from 0, Set the gamma to 1.2 from 1.0 and set the white point to 245 from 255. Then I look at the Action Manager code Adobe's plug-in scriptlistener recorder.  I then used the Clear SL Photoshop Script to make the code more readable.  Then create a javascript Function for Levels that may work  in your Photoshop Script.  I have no idea how you would know the white point the script should set. To make an auto white point script.

Action Managet Code:

// =======================================================
var idhistoryStateChanged = stringIDToTypeID( "historyStateChanged" );
    var desc1440 = new ActionDescriptor();
    var idDocI = charIDToTypeID( "DocI" );
    desc1440.putInteger( idDocI, 1889 );
    var idIdnt = charIDToTypeID( "Idnt" );
    desc1440.putInteger( idIdnt, 1894 );
    var idNm = charIDToTypeID( "Nm  " );
    desc1440.putString( idNm, """Levels""" );
    var idhasEnglish = stringIDToTypeID( "hasEnglish" );
    desc1440.putBoolean( idhasEnglish, true );
    var idItmI = charIDToTypeID( "ItmI" );
    desc1440.putInteger( idItmI, 3 );
executeAction( idhistoryStateChanged, desc1440, DialogModes.NO );

// =======================================================
var idLvls = charIDToTypeID( "Lvls" );
    var desc1441 = new ActionDescriptor();
    var idpresetKind = stringIDToTypeID( "presetKind" );
    var idpresetKindType = stringIDToTypeID( "presetKindType" );
    var idpresetKindCustom = stringIDToTypeID( "presetKindCustom" );
    desc1441.putEnumerated( idpresetKind, idpresetKindType, idpresetKindCustom );
    var idAdjs = charIDToTypeID( "Adjs" );
        var list19 = new ActionList();
            var desc1442 = new ActionDescriptor();
            var idChnl = charIDToTypeID( "Chnl" );
                var ref113 = new ActionReference();
                var idChnl = charIDToTypeID( "Chnl" );
                var idChnl = charIDToTypeID( "Chnl" );
                var idCmps = charIDToTypeID( "Cmps" );
                ref113.putEnumerated( idChnl, idChnl, idCmps );
            desc1442.putReference( idChnl, ref113 );
            var idInpt = charIDToTypeID( "Inpt" );
                var list20 = new ActionList();
                list20.putInteger( 20 );
                list20.putInteger( 245 );
            desc1442.putList( idInpt, list20 );
            var idGmm = charIDToTypeID( "Gmm " );
            desc1442.putDouble( idGmm, 1.200000 );
        var idLvlA = charIDToTypeID( "LvlA" );
        list19.putObject( idLvlA, desc1442 );
    desc1441.putList( idAdjs, list19 );
executeAction( idLvls, desc1441, DialogModes.NO );

Photoshop Clean SL script code:

// =======================================================
historyStateChanged(1894, "Levels", true, 3);
function historyStateChanged(ID, name2, hasEnglish, itemIndex) {
	var descriptor = new ActionDescriptor();

	descriptor.putInteger( stringIDToTypeID( "documentID" ), 1889 );
	descriptor.putInteger( stringIDToTypeID( "ID" ), ID );
	descriptor.putString( stringIDToTypeID( "name" ), name2 );
	descriptor.putBoolean( stringIDToTypeID( "hasEnglish" ), hasEnglish );
	descriptor.putInteger( stringIDToTypeID( "itemIndex" ), itemIndex );
	executeAction( stringIDToTypeID( "historyStateChanged" ), descriptor, DialogModes.NO );
}

// =======================================================
levels(1.2);
function levels(gamma) {
	var descriptor = new ActionDescriptor();
	var descriptor2 = new ActionDescriptor();
	var list = new ActionList();
	var list2 = new ActionList();
	var reference = new ActionReference();

	descriptor.putEnumerated( stringIDToTypeID( "presetKind" ), stringIDToTypeID( "presetKindType" ), stringIDToTypeID( "presetKindCustom" ));
	reference.putEnumerated( stringIDToTypeID( "channel" ), stringIDToTypeID( "channel" ), stringIDToTypeID( "composite" ));
	descriptor2.putReference( stringIDToTypeID( "channel" ), reference );
	list2.putInteger( 20 );
	list2.putInteger( 245 );
	descriptor2.putList( stringIDToTypeID( "input" ), list2 );
	descriptor2.putDouble( stringIDToTypeID( "gamma" ), gamma );
	list.putObject( stringIDToTypeID( "levelsAdjustment" ), descriptor2 );
	descriptor.putList( stringIDToTypeID( "adjustment" ), list );
	executeAction( stringIDToTypeID( "levels" ), descriptor, DialogModes.NO );
}

A Function that may work in youe Photoshop script:

levels(0,1.0,255);   // default no adjustment Black 0, Gamma 1, White 255

function levels(black,gamma,white) {
	var descriptor = new ActionDescriptor();
	var descriptor2 = new ActionDescriptor();
	var list = new ActionList();
	var list2 = new ActionList();
	var reference = new ActionReference();

	descriptor.putEnumerated( stringIDToTypeID( "presetKind" ), stringIDToTypeID( "presetKindType" ), stringIDToTypeID( "presetKindCustom" ));
	reference.putEnumerated( stringIDToTypeID( "channel" ), stringIDToTypeID( "channel" ), stringIDToTypeID( "composite" ));
	descriptor2.putReference( stringIDToTypeID( "channel" ), reference );
	list2.putInteger( black );
	list2.putInteger( white );
	descriptor2.putList( stringIDToTypeID( "input" ), list2 );
	descriptor2.putDouble( stringIDToTypeID( "gamma" ), gamma );
	list.putObject( stringIDToTypeID( "levelsAdjustment" ), descriptor2 );
	descriptor.putList( stringIDToTypeID( "adjustment" ), list );
	executeAction( stringIDToTypeID( "levels" ), descriptor, DialogModes.NO );
}