This pretty much comes down to microphone polar patterns, you can read more about it here: Polar Patterns.
Also, condenser microphones are more sensitive than dynamic microphones. You can read more about it here: Dynamic vs Condenser. Your lavalier microphone is likely a condenser.
A dynamic microphone with a cardioid pickup pattern in a configuration where the rejection point of the microphone is facing the TV would be an optimal solution. These are the same kinds of microphones that you see vocalists use on stage. A shure sm58 or an sm57 are prime examples, but there are cheaper brands (like behringer) that make similar models at a lower price point.
If you go this route, you will also need a way to connect the microphone to you computer, which is typically done with an audio interface. The problem is, between this and the microphone, I can almost guarantee you will be over budget. Because of that, I'm going to suggest sticking to USB microphones.
Blue Microphones makes some great stuff tailored towards the gaming and podcasting communities (as well as some very nice professional capsule mics). You should check out the snowball and yeti models. Although they are condenser mics, they do have cardioid polar pattern options, and you should be able to get pretty good results if you keep the gain low and eat the mic (keep it close to your mouth) while attempting to keep the TV in the rejection areas.
Using a software noise gate may help as well (free ones exist). This essentially mutes your mic until the input signal reaches a certain volume, which means your friends would only hear the TV while you are talking. If you use Discord for your chat software (most gamers do these days), I believe they have a noise gate built-in that you can enable through settings.
Hope this gives you a starting point. Good luck!