What is wrong with allowing competition with the Fed?
In so far as the US government is going to back a currency all Congress needs to do is set the standard -- i.e., tie a currency to some fixed quantity of good; for example, 1 ounce of gold equal to some paper equivalent. There is nothing else Congress needs to do.
However, I don't think they should even be allowed to do that. Let the markets figure out what consumers want and government spending be damned.
Those who favor limited government should also consistently favor a hard money standard. That is the only thing that can reign in government growth and debts.