Script for manipulating images in a folder and saving with background watermark

You would add the watermark/logo  as the top  layer in the layer stack  so it would be visible in the saved image file. You would use the script in an action that you would batch with menu file>Automate>Batch... or menu File>Scripts>Image Processor or menu File>Automate>Image Processor Pro.   The Script I created  I have posted. "PlaceLogo.jsx" You could use that script customized to use your watermark. You just need toe edit a few var statement.

If you want to resize the png images to square 1024x1024 I would use Fit resize the png images to fit some area like 1000px by 1000px then add canvas  with canvas size 1024px by 1024  with the anchor point centered,  You do not want to distort the png  image content. The Watermark script step should be after the canvas has been resized to  a 1024x1024 canvas. Saving as a jpeg will add a solid background color to any transparent pixel Jpeg does not support transparency.

/* ==========================================================
// 2021  John J. McAssey (JJMack) 
// ======================================================= */
// This script is supplied as is. It is provided as freeware. 
// The author accepts no liability for any problems arising from its use.
/*
<javascriptresource>
<about>$$$/JavaScripts/PlaceLogo/About=JJMack's PlaceLogo ^r^rCopyright 2021 Mouseprints.net^r^rPhotoshop Script^rCustomize using first four var</about>
<category>JJMack's Script</category>
<enableinfo>true</enableinfo>
</javascriptresource>
*/
#target photoshop;  
app.bringToFront();  
var logoFile = "~/Desktop/JJMack.png";				// Logo Image file should be large for resize down works better than up. Vector files can be any size.
var LogoSize = 10;											// percent of document height to resize Logo to
var LogoMargin = 1;                                         // percent of Document height the Logo should have as a margin
var LogoLocation = 9;										// Like a tick tack toe board 1 through 9.  1=Top Left 9=Bottom Right
var LogoRotation = -20;										// Center Logo Location 5 can be Rotated -90 through +90 degrees  + degree Clockwise - CounterClockwise

app.displayDialogs = DialogModes.NO;				// Dialog off 
var strtRulerUnits = app.preferences.rulerUnits;	// Save Users ruler units 
var strtTypeUnits = app.preferences.typeUnits;		// Save Users Type units 
app.preferences.rulerUnits = Units.PIXELS;			// work with pixels 
app.preferences.typeUnits = TypeUnits.PIXELS;		// work with pixels 
//placeLogo(logoFile, LogoSize, LogoMargin, LogoLocation, LogoRotation); // Place Logo into the location 5 would be Document's center
if (documents.length) app.activeDocument.suspendHistory('placeLogo','placeLogo(logoFile,LogoSize,LogoMargin, LogoLocation, LogoRotation)' );
app.preferences.rulerUnits = strtRulerUnits;		// Restore user ruler units  
app.preferences.typeUnits = strtTypeUnits;			// Restore user type units    
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
function placeLogo(Image,Size,Margin,position,angle){  
	if(!documents.length) return;  							// if no document return
	var fileObj = new File(Image);	 		               	// the passed file
	if(!fileObj.exists){  									// If file does not exits tell user 
		alert(fileObj.name  + " does not exist!"); 			// Alert User 
		return;  											// return
	}  
	try{  
		var doc = app.activeDocument;						// set Doc object to active document
		var layers = app.activeDocument.layers;				// get layers
		app.activeDocument.activeLayer = layers[0];			// Target Top Layer
		placeFile(fileObj); 								// Place in file the Logo File
		activeDocument.activeLayer.resize(100 ,100,AnchorPosition.MIDDLECENTER); // Insure Place did not scale layer  
		var SB = activeDocument.activeLayer.bounds; 		// get layers bounds 
		var layerHeight = SB[3] - SB[1];					// get layers height  
		var resizePercent = (100/layerHeight)*(Size/100*doc.height.value); // Percent to resize by 
		activeDocument.activeLayer.resize(resizePercent ,resizePercent,AnchorPosition.MIDDLECENTER);  // Resize width and height by percentage 
		marginSize = Margin/100*doc.height.value;			// Margin size
		var selectedRegion = Array(Array(0+marginSize,0+marginSize), Array(doc.width-marginSize,0+marginSize), Array(doc.width-marginSize,doc.height-marginSize), Array(0+marginSize,doc.height-marginSize));
		activeDocument.selection.select(selectedRegion);    // Select  document area for the logo Positioning
		switch (position){									// Align resized logo smart object layer into position
			case 1 : align('AdLf'); align('AdTp'); break;
			case 2 : align('AdCH'); align('AdTp'); break;
			case 3 : align('AdRg'); align('AdTp'); break;
			case 4 : align('AdLf'); align('AdCV'); break;
			case 5 : align('AdCH'); align('AdCV'); activeDocument.selection.deselect(); activeDocument.activeLayer.rotate(angle); break;
			case 6 : align('AdRg'); align('AdCV'); break;
			case 7 : align('AdLf'); align('AdBt'); break;
			case 8 : align('AdCH'); align('AdBt'); break;
			case 9 : align('AdRg'); align('AdBt'); break;
			default : break;
		}
		app.activeDocument.selection.deselect();			// deselect
	}
	catch(e) { alert(e + ': on line ' + e.line); }			// inform user of error  
	finally{ }  
} 
//////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// 
function placeFile(placeFile) {  
    var desc21 = new ActionDescriptor();  
    desc21.putPath( charIDToTypeID('null'), new File(placeFile) );  
    desc21.putEnumerated( charIDToTypeID('FTcs'), charIDToTypeID('QCSt'), charIDToTypeID('Qcsa') );  
    var desc22 = new ActionDescriptor();  
    desc22.putUnitDouble( charIDToTypeID('Hrzn'), charIDToTypeID('#Pxl'), 0.000000 );  
    desc22.putUnitDouble( charIDToTypeID('Vrtc'), charIDToTypeID('#Pxl'), 0.000000 );  
    desc21.putObject( charIDToTypeID('Ofst'), charIDToTypeID('Ofst'), desc22 );  
    executeAction( charIDToTypeID('Plc '), desc21, DialogModes.NO );  
} 

function align(method) {
	var desc = new ActionDescriptor();
	var ref = new ActionReference();
	ref.putEnumerated( charIDToTypeID( "Lyr " ), charIDToTypeID( "Ordn" ), charIDToTypeID( "Trgt" ) );
	desc.putReference( charIDToTypeID( "null" ), ref );
	desc.putEnumerated( charIDToTypeID( "Usng" ), charIDToTypeID( "ADSt" ), charIDToTypeID( method ) );
	try{
		executeAction( charIDToTypeID( "Algn" ), desc, DialogModes.NO );
	}catch(e){}
}

Use Image Processor/Image Processor Pro to resize your PNG files first, then you can use a script or data-driven graphics to save out the composites.