Yeah but you still didn't answer the question, how do you know if you have audio signal if the speakers cut out, it doesn't matter if the BT is still working.
Here's my consensus, your factory stereo use a floating ground setup, that means there is no common ground, you didn't mention using a floating ground adapter so i must assume you didn't, in essence you hooked up a stereo that uses a common ground to a speaker array that uses a floating ground, the output transistors are getting smoking hot and shutting down, the fact they recover and begin working again after a period of time is amazing but lays testament to the quality of the output transistors. An easy to know if your system uses a floating ground (as an example) is if you have 8 speakers but only 5 speaker wires, if it had a common ground you would have 8 speakers and 16 speaker wires.
To begin with the stereo is cooked, it will eventually fail and not recover as the output transistors sustained damage. You can use a floating ground adapter but they will throw your balance and fader control into total disarray and cut the sound output by about 40%, you'll have to turn the volume way up to get any audio and it will sound like garbage.
You can only use 4 speakers with your aftermarket head unit and you have to remove all 4 of them to rewire them for use with a common ground, they are crappy speakers and you are better off replacing them with some quality 5x7s which are already designed to use a common ground.
While the stereo and speakers are out you will have to run new speaker wires from the head unit to each speaker, 1 pair of wires per speaker.
Or, you can use a floating ground adapter, your choice..
Don't forget to check for correct polarity on the speaker terminals, sometimes they get the + and - mixed up and the motor pulls the cone in when it should be pushing out and vice versa..
My personal audio system is a 5-channel full-active setup pushing 800 watts at full boogie, you know you did it right when your pumping out the tunes and people roll their windows down instead of up
On the same note, I could be dead wrong about everything and the problem lies elsewhere but that's my best educated guess since the problem didn't exist until you reolaced the head unit..