My solution: I have online ATC on my headset by default. When doing long FS2crew interaction such as airborn briefings I simply go to speaker. 

That guarantees that I am able to hear what ATC is saying. 

 

If that's too complicated, you could also complete the briefing and ask te controller to repeat what he was saying. Happens in the real world all

the time!

In the next update, there's a key command you can use to pause the entire FS2Crew audio engine.

 

You'll see it first in the Majestic Dash 8 update, which I may release next week along with the Dash 8 Global FO voice set.