Social Security & Medicare

I'm a lawyer but these matters are largely governed by state based laws, and I know next to squat about food stamps/welfare/medicare law anyway, except as it relates to child support which I doubt he pays.

You should find out first whether your knowledge of his lying, may get you in trouble. I doubt it but check.

You definitely avoid helping him commit fraud. If the school asks for his address, you give the correct one.

What if you won't get in legal trouble by being aware of but not disclosing his lying?
Up to you.

You could tell him you can't and won't cover for his lies because that gets you in serious trouble, and its immoral, so he needs to seriously consider whether his actions will somehow force you to disclose information that gets him in trouble.

His lies seem like they can be easily caught, and would amount to fraud.
All a social agency needs to open an investigation to find fraud is a report from a school, court, tax authority, other government agency, check the family court and see the shared custody order, etc etc.

Welfare fraud is usually a felony punishable by fine and possible imprisonment.

A lawyer may keep him out of jail but he'll have to pay everything back, and could have a felony on his record he would need to disclose on future food stamp, medicare, disability applications, job applications, maybe credit card/debt applications, etc.

Hopefully he'll get scared enough to withdraw the application.