How to trick smart TV into thinking my router is connected to the internet?

I have a PC setup with lamp if that helps, but I'm still learning.

I am attempting to screen mirror my phone to TV and for whatever reason this seems to be designed to not work unless there's a router setup as sort of a middle man between the phone and TV. (WiFi direct doesn't seem to be a mainstream option for now.)

I'd like to purchase a router, connect phone and TV to it, and then screen share my phone. One problem is that the TV refuses a WiFi connection if there is no internet connection present on the wifi.

Is there some way to have a router connect to a lan, so that the tv could ping the lan and accept the connect to WiFi?