Playing with this some more and discovered with the modern player, if you want the Print button to consistently appear to the left of the screen (e.g. As it does with Math's story file example; near the volume button) then you need to add this script each time to each slide (trigger executes was slide/layer starts) where you want the button to appear.
Maybe there is another way, but I tried everything and it wasn't until I put this on each slide did behave as I had hoped. Otherwise the button pops up almost to the right of the player.
var printButton = document.querySelector("#link-right-0");
var volumeButton = document.querySelector("#volume")
volumeButton.after(printButton);
printButton.className= "btn cs-button volume-panel";
gsap.set(printButton, {
height:'30px',
x: '80px',
y: '8px',
zIndex: 1000,
borderColor: "#B1B1B1",
autoAlpha:0
});