To get exactly what you'd want you'd have to do a lot of work.
Here is my suggested, simpler answer:
Don't open up a new tab. If you use showPage like you mention, and provide a "back" button on the page to go back to where you were, you'll get pretty much everything you need. If you don't want the back to show up when you print, then you can setVisibility(false) on the button before you print, then print, then setVisibility(true).
I'll give a quick summary of how you could do this with a new tab, but it's pretty involved so I can't go into details without trying it myself. The basic idea, is you want to open the page with a full URL, just like a user was navigating to it.
You can use #TestPrint to indicate which page you want to load. You also need the URL of your application, which as far as I can remember is only available in a server-side script using the Apps Script method: ScriptApp.getService().getUrl(). On top of this, you'll probably need to pass in the key so that your page knows what data to load.
So given this, you need to assemble a url by calling a server script, then appending the key property to it. In the end you want a url something like:
https://www.script.google.com/yourappaddress#TestPage?key=keyOfYourModel.
Then on TestPage you need to read the key, and load data for that key. (You can read the key using google.script.url).
Alternatively, I think there are some tricks you can play by opening a blank window and then writing directly to its DOM, but I've never tried that, and since Apps Script runs inside an iframe I'm not sure if it's possible. If I get a chance I'll play with it and update this answer, but for your own reference you could look here: create html page and print to new tab in javascript
I'm imagining something like that, except that your page an write it's html content. Something like:
var winPrint = window.open('', '_blank', 'left=0,top=0,width=800,height=600,toolbar=0,scrollbars=0,status=0');
winPrint.document.write(app.pages.TestPage.getElement().innerHTML);
winPrint.document.close();
winPrint.focus();
winPrint.print();
winPrint.close();
Hope one of those three options helps :)