When I'm connected to Android Auto wirelessly and a call comes in, I answer it and if I need to reroute the audio to my headset, I press the Bluetooth icon on the display and it gives me the choice to switch it to handset, when I press handset, the audio gets routed to my phone handset for privacy. Do you guys not have that option on your Android Auto screen???

 

That's it! I just found out that I needed to go into ( on my phone) settings, apps, android auto, permissions and add phone, contacts. I had permissions set in the phone only. Once I closed android auto them shut off truck completely and restarted I got the screen you just showed. This is exactly what I was looking for. If I switch to phone it stays there the duration of car cycle.
thanks

 

Just came here to post the same thing.
Can't listen to my blue tooth ear pod from my Samsung S23+. No matter what I try it switches to the car audio in the 2022 Lariat. I think the only way it would work is to disable Android play and disconnect the blue tooth from the car.
I didn't have that problem in any of my other Fords with the same phone. 13 F150 Limited, 18 Edge, 21 Explorer
All of those will switch from the car to the Ear pods with just turning them on. And back without much effort.