PS4 – picture on beamer, audio on Apple TV 4K – how?

If the screen/projector doesn’t have ARC/eARC, then you can only use the HomePod Mini speakers for content originating on Apple TV 4K (or AirPlay of audio-only sources). That is what these devices offer. That is the method used to bring this feature.

Use HDMI ARC or eARC with your Apple TV 4K (2nd generation) - Apple Support

What you need

• Apple TV 4K (2nd generation) [or later*]

• HomePod mini or HomePod speakers

• A TV that supports HDMI ARC or eARC

Notice the third point.

Can the sound be transmitted from the PS4 to the Apple TV 4K directly, without taking the detour via the projector?

I am pretty sure that is not supported. Only ARC allows for two-way signaling over HDMI. Full use of video and audio one way, and audio only in the return direction.

Toslink to HDMI adapter […] Is there such a thing?

I have never seen such a device for that direction. Adapter are directional, so HDMI to Toslink wouldn’t help. I think you will need a different speaker system for your PS4–beamer setup.

The intended use of that device is different. It is suppose to bring eARC bandwidth (e.g. Dolby Atmos) to users that only have ARC bandwidth (Dolby 5.1 or lower) on their TV, so those users don’t have to limit themselves by using audio passthrough on the TV. It should get its input from the player device, and output to screen and audio system separately with two HDMI outputs.

A bit pricy for unknown if it will work.

Could that work?

Not sure. I have never read user stories about such use with this device. You could try contacting the manufacturer.