Since CLEO won't sign off, why not shift it over to the state police? Louisiana is doing it for form 4's(suppressor, short-barreled shotgun, or short-barreled rifle), but not Form 1's or full auto.
Sheriff is not the only CLEO who can sign, anyway - state police barracks captains, district attorneys, US attorneys, judges and a host of other officials can also sign for individual NFA forms.