OK, my latest information is *no* different from before, unfortunately.
First, once I have the Dell disconnected, there is no way to switch the menu bar to the other screen. But I did it anyway before disconnecting.
So here I am with my MBP (built-in display only) and my TV in front of me. I click the Airplay icon on my menu bar and choose the only choice: Airplay to: ATV. My MBP monitor loses the menus and only shows the background, and the TV shows what used to be on my MBP monitor - all the windows and menus. Good right? Now... how do I choose what to see on the TV? Right now it is just mirroring (though I didn't choose to have it mirror... not a menu option... and no more menu on my MBP). I can *sometimes* move the cursor on the TV with my touchpad. But other times it disappears. And when I try to click something, only a few things respond. And if I move the cursor up to the Airpaly icon in the menu bar (on my TV) and click, it then freezes. And now pressing menu on my ATV remote doesn't do anything. I sit there for a minute or so, and everything times out, and control is restored to my MPB, and the TV just shows the ATV menu.
1. How do I choose what to see on my TV? (If I start something playing on the computer, and THEN click the Airplay menu, that will then play on the TV. But then I have to go back to the beginning with the ATV remove and start the video from the beginning. This can't be right.)
2. Is this really how it is supposed to work, and I'm expecting too much?
3. this all worked correctly the last time I tried it many months ago, so I don't think #2 is accurate. What the heck may have changed?
The good news? As long as I don't try to click the Airplay menu on my TV, I DO have a clean way of getting out - by using the menu button on the ATV remote as you suggest. So that part is great! Now I just need to figure out how to make the Airplay work for me!