Ok.. so you have limited choices. The tape deck and PC can ONLY be plugged into the PC input. You tried the game audio in and that did not work. So now you either experiment with buying a bluetooth setup (which I would hesitate doing esp since it has to travel to another room) or also buy a splitter type of device that you can manually switch between PC and tape deck for the input to the Sony.

something like this..

https://www.amazon.ca/ZIEM-LINEPAUDI.../dp/B09BCPDZ2X

which can be used as a two in, one out stereo device. Note that you'd still need the 3.5 to rca stereo adapter unless you could find a splitter with a stereo RCA and a stereo 3.5 input to a stereo RCA output but that's getting a little esoteric.