I think the questions they need to ask themselves are these:
Is it pleasing to God to fornicate and not really be married?
OR
Is it pleasing to God to lie to, cheat, or defraud the people by pretending not to be married when they really are?
I understand some people don't want to register their marriages for fear of big brother. But here, the motivation seems to be to get money when they aren't legally entitled to it. Maybe the motivation for that is to be able to eat, so I understand.
I think there is something to be said for just doing the brave thing and trusting God to provide some way if they want to be married. In some states, they might be considered legally married even if they don't report it, so they could get in trouble later on.
This is a big problem in the US, and I wish the government would change laws to be more friendly to seniors getting married without having financial incentives against them for doing so. When even the older people start shacking up, that's got to have a negative, destabilizing effect on family in society.