However, you will never eliminate welfare payments 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.
You are thinking only in economic terms. I think it would be a better legal and moral system. By registration of the marriages the people involved would then be traceable. Not a perfect system but probably better. We have special laws on the books for special groups already; i.e. as a white try to get a government job! Apply to Wells Fargo bank! Apply to medical school as a white male. Become a provider to the Federal Government.