When you do Automate -> Batch, you refer to a certain script (or "action" in the Photoshop parlance). Just make that action explicitly save the image the way you want.
Open a sample image.
Create a new action and start recording.
If you want to export in a certain profile, do Edit -> Convert to Profile explicitly.
Do File -> Save As...,...
Mostly it is about color temperature, white balance (they're not the exact same thing), contrast, and then using an HSL/HSV/HSB (Hue-Saturation-Luminance/Value/Brightness) tool to mute some colors, shift the hues of other colors, and emphasize other colors.
Choose "fixed size" in the Marquee selection tool ribbon and set it to 720x720 (or 720xH for whatever your height is). Then place a selection on the canvas. Then drag it left or right until it centers (the center vertical line should show when it does).
Another potential way to do this is by changing the canvas size. (I don't have Photoshop, but I believe that this will work - it works in Affinity Photo :D)
Select Image=>Canvas Size (which opens the Canvas size dialog box)
In the "New Size" section, set the anchor to be the center box
Leave the "Height" as is
Set the "Width&...
Turn on the rulers. In my version there is a selection under the View menu.
Make sure guides are visible under View->Show
Drag from the vertical ruler on to the picture to create a guide. As you move it you can see the pixel position. Position guides at the points you want to select. You may have to zoom in a lot to get the exact pixel position.