Xbox communication suspension by my bratty cousin

Unfortunately, the Xbox Policy & Enforcement Team have no way to say it was you or your cousin who conducted the behavior that earned the communication ban. It'd be easy to accuse someone else of doing it to get lifted which is why they can't really
take into consideration and only look at the evidence, the profile who the action is being held against. If your appeal already been used up, by the sounds of it, you wont be able to try and get it lifted.

As far as your gaming tournament goes, you might want to see if you team can't use an outside voice system, such as Discord in the meantime. This is an app that can be used on your phones.

My recommendation, in the future turn off the auto-login so that even if he finds the controller again you don't end up getting a permanent ban for repeated offenses. I would also bring it up with your family and might even consider taking away the power
cords to the box in the future so they can't even touch it. (I've done that before)

Im sure you can appeal a case over with the enforcement team, that can however say that whoever was on your account, that it wont matter and you hold all responsibility for your account no matter whos on it.

You are responsible for whatever happens in your name (account)

they already did that, and their mother is an entitled person so it is MY fault for some reason

As the Console owner and account owner you are responsable for anything that happens on your account. You can try rasing a case but i doubt it will work as everyone than claim it was not them.