Only play the clip when you are ready.

Flash block {web link} Never be annoyed by a Flash animation again! Blocks Flash so it won't get in your way, but if you want to see it, just click on it

I don't have a personal recommendation.

I spent a few minutes searching the Add-ons site and it seemed that Noise Control was the best rated/most developed, but I didn't check it out personally to see whether it has automatic muting of background tabs.

https://addons.mozilla.org/firefox/addon/noise-control/

If you try it, let us know whether it works.

Oh, actually I see from your Question Details > More System Details that you already have Noise Control. I guess that means it doesn't do what you need. Have you found its sidebar feature?

Noise Control only works for HTML5 vids

Sidebar feature?

The version notes for Noise Control says it has a sidebar you can open to see where all audio is playing and, I assume, centrally manage it.

But if it doesn't monitor plugin-generated audio, it's obviously not a complete solution.