Absolutely there should be no welfare payments at all. However, you will never eliminate welfare payments now. Too late! Experience shows once started a state cannot stop i.e.tThe Romans were still on the dole as Alaric was beating down the city gates. Welfare will only expand. I expect the government to provide “Free” tickets to various sporting events at anytime now.
Given that stipulation, why do you imagine you could put in place a unique set of rules applying only to legally-recognized polygamous families? Even on the snowball's chance you passed it, even at a county level, a judge would throw it out before you could pat yourself on the back.
We already have religiously-recognized polygamous "households" on welfare, so there's no percentage for them in wanting a legal marriage, if that would threaten the income.