I've tried all on these except buying another pc ..
option c) Check out protondb to see other users experience with game performance on Steam deck. Sadly it looks like Rocket League has just lost native Linux support. https://www.protondb.com/app/252950
Option D) I have network cables running from pc upstairs to living room (just clipped to the skirting boards and run under rugs) and have used a Pi4 with steam link/moonlight and it works fine. There are some compression artifacts but overall it's hardly noticeable and quite playable.
Option E) Another way was using my Samsung tablet or phone with a usb to hdmi adapter which will mirror the game on the TV and I've used steam link that way (with controller connected to tablet via Bluetooth)
Option A) And just finally to note I also have a TV in the office with the PC and use a 4m long hdmi cable with that sometimes too which works perfectly, though it was hard to find a good quality hdmi cable that worked.
That fake second monitor is a fantastic idea, didn't even know that was a thing!