DP out to usb-c to hdmi. Is that possible?

I want to connect my desktop computer to the TV through a hole in the wall which should be as small as possible. The two devices are only couple of centimeters away from each other, so cable length should not matter.

My first idea was to connect the PC via an HDMI-to-USB-C cable and then go back to HDMI again. Problem is that I could not find an HDMI-to-USB-C cable that transports the signal comming from HDMI to USB-C but just the other way round.

So I’ve come up with the idea to use a DisplayPort-to-USB-C cable which is supposed to be biderectional, plug the DP end to the PC, connect the USB-C end via an USB-C coupler tho the USB-C-to-HDMI cable and plug the HDMI end to the TV.

I’ve got a HDMI to USB-C cable which works totally fine if I connect a laptop to my TV for instance. Same with the DP-to-USB-C cable (of course in the “wrong” direction). What I cannot test in my setup is the USB-C coupler and weather the DP-to-USB-C cable really is bidirectional.

My question now is if this setup, in theory, is possible in the first place?