and I will star this by noting that I am NOT a lawyer. I have lived in the DC area for 37+ years and have some understanding of the roles that various law enforcement and military units can play.
The President has complete authority over any Federal land — thus using Military Police in Lafayette Park is completely within his authority.
The mounted police unit that cleared H Street next to Lafayette Park were Park Police, who have some police authority any place in the Metropolitan region, although usually to make an arrest they are supposed to get clearance from at least a Sergeant. I am presuming that the command structure of the Park Police did authorize the use of the mounted unit.
Secret Service have the authority to arrest, to secure, and if necessary to clear areas to ensure the safety of their protectees, obviously including the President.