The US President is Commander-in-chief of the US military.
The President shall be commander in chief of the Army and Navy of the United States, and of the militia of the several states, when called into the actual service of the United States;
(from Article II section 2)
That does not make the president the direct boss of every federal employee. The Congressional Sergeants-at-arms, in particular (and their assistants) are employed by, and responsible to, Congress, not the President. The Secret Service is part of the Department of Homeland Security (formerly part of the Treasury Department, until 2002) which is part of the Executive branch, but I am sure the President cannot order them to arrest someone who has not committed any crime. If such a thing were pushed to a direct confrontation, I have no idea where it would go, I hope we do not find out.
Article I Section 8 grants Congress the power (among a number of others):
To exercise exclusive legislation in all cases whatsoever, over such District (not exceeding ten miles square) as may, by cession of particular states, and the acceptance of Congress, become the seat of the government of the United States, and to exercise like authority over all places purchased by the consent of the legislature of the state in which the same shall be, for the erection of forts, magazines, arsenals, dockyards, and other needful buildings
That seems to say the ultimately Congress controls the District, and sets the rules there. There is also the provision in Article I section two that:
The House of Representatives shall choose their speaker and other officers;
which would include the Sargent-at-Arms, I think.
Article I section five says:
Each House may determine the rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two thirds, expel a member.
which again seems to grant control over the situation to the individual houses of Congress.
Article I section 6 says:
The Senators and Representatives shall receive a compensation for their services, to be ascertained by law, and paid out of the treasury of the United States. They shall in all cases, except treason, felony and breach of the peace, be privileged from arrest during their attendance at the session of their respective Houses, and in going to and returning from the same; and for any speech or debate in either House, they shall not be questioned in any other place.
which again puts Congress out of the direct control of the President.