How can I set a custom menu shortcut for Mark as Played in the macOS Podcasts app?

In the experiments that I've tried, this is not possible.

As I said in a comment above, I didn't even know that you could assign a shortcut to a contextual menu item. I tried it, and I can get it to appear as in your pic, and I can assign a VERY unique key combo to it. But the key-combo just doesn't work. I even tried with another contextual menu item:

It's not visible in my pic, but that key-combo is also greyed like your F1 example. I moved this key-combo to another "real" menu item (Half Speed), and it works there.

My conclusion: contextual menu items can't really be assigned keyboard shortcuts. I tested in the TV app as well, same results.

I even went a bit farther, looking in Automator and the AppleScript, and there are no "dictionaries" for scripting the Podcast app.

I'm afraid this isn't possible (yet) with this app.