Unequivocally true in Michigan and here in Colorado, and likely true in most states that recognize the Michigan permit.
But the question didn't specify the location or if the BUG is to be concealed. The Michigan permit is recognized in New Mexico, but one must play by local rules which authorize only 1 concealed firearm. So if the implication was "can I carry a concealed primary and concealed BUG?", the answer would be "no" in New Mexico. You could carry a primary and BUG there, but one of them must be openly carried.