So you only have a picture via a projector and when you connect it to the AppleTV 4K, you have a picture as standard?
Then this Delock is certainly worth a try!
It's just a question of what's smarter, audio cable jack so something like this: ready2music R2MAUX (1.20m, 3.5mm jack, cinch) or TOSLINK cable and with TOSLINK > jack 3.5mm adapter