You can do it. but your answer is in the words below.
The difference between the two BCGs is the presence of a full auto lug. The bottom portion of the BCG has a lug that contacts the sear release in a full auto trigger groups and allows the gun to fire in fully automatic. Since a standard AR 15 does not have the trigger components to fire full auto the presence of the full auto lug does not allow a standard AR 15 to fire full auto.