When you say the TV is the only source of light on that side of the room, do you mean that you want it actually lighting/exposing the people and the room or is it just a bright thing in the corner of the room? Is there no possible other sources as well?
When the TV is a light source, I wouldn't green screen it, it should be bright, a bit overexposed and that would add some natural effect on whatever is right next to the TV. I'd only green screen a TV if it was in brighter room.
TV's don't really flicker much, it's like a scene in a movie theater, the brightness changes as the image changes or when cuts happen to shots that are brighter or darker. If it has to exactly tie into the action and cutting on the TV, then it's a bit tricky, you might have to get a video projector with the same feed as the TV set and use it as a light, maybe through a light diffusion frame, and figure out how to get it off-camera but shining from a similar direction as the TV set.
Otherwise if the flicker / shifting could be done randomly, with subtlety, from off-camera and not be too exactly the same as the footage on the TV set, there are any number of ways to do that, just like when doing a firelight gag. Could just be an LED soft light, for example, slightly fading up and down randomly by turning the dimmer knob. Or two soft lights, one fading up and down a little differently than the other. I've done things like put two or three small lights through one diffusion frame, one light flickering, one light just dimming up and down, and maybe a third light steady. I've also just waved my hands or a small flag randomly between a light and a diffusion frame. You have to have a feeling for what the effect would be in real life, like a sudden fast shift flicker for a scene cut in the footage and some side to side motion for brightness changes moving across the screen. It has to feel organic and random, and it's better to be subtle unless you want a stylized strobing effect for a horror moment.