Not to be Captain Bringdown in this scenario, but the actual battle would not be between USMS and USSS, but between the White House Counsel and President Trump's personal attormey, and on the other side the staff attorneys for (I would guess) the House Judiciary Council and probably the U.S. Attorney for the District of Columbia. They would hash it out and on advice of counsel, would decide whether the President would be subject to arrest, and certainly conduct iiaison with the Director of the USSS before any arrest. The Director of the USSS, like all federal officers, made an oath to the Constitution and the lawful orders of the President, it's not like the USSS is a Praetorian Guard who has sworn a blood oath of allegiance to the President. If there was a lawful warrant, they would be obligated to assist in the legal process. Trump has actually shown a record of complying with congressional and judicial actions to date, it's not like he would refuse to comply I doubt gunplay would be involved, although it's fun to imagine this as an episode of that old show "Deadliest Warrior" (like, "who would win if al Qaeda fought the IRA?")
You are not going to put the President of the United States in a D.C. jail for obvious reasons, however much Nancy Pelosi might want a perp walk. They would probably bring a magistrate in to the Oval Office, make a determination as to what happens next, and release him on his own recognizance. A determination would be made by the President and VP as to whether the VP wpuld assume the office in this case.
The USMS's jurisdiction lies (or at least this used to be the case, Muay Tired can tell you for sure) with those whose cases have previously been before a magistrate, whether state or federal...so Trump woud not be subject to arrest by the Marshals. (Generally speaking the FBI handles fugitives who have not yet been adjudicated as UFAPs, Unlawful Flight to Avoid Prosecution.) Depending on which crime of which Trump would be charged, the federal law enforcement branch which has jurisdiction over that violation would make the arrest. Each agency serves its own warrants. So if Trump was charged with, say, an IRS violation, the IRS would serve the warrrant. As the FBI has jurisdiction over the greatest number of federal violations, most of which would be the ones likely to be charged in this scenario, the FBI would likely be the ones to serve the actual warrant.
The closest thing to such a scenario that has actually happened was Watergate, probably. One of the senior agents on my squad when I first started working was assigned to the Washington D.C. Field Office when he was a young agent and had to serve a warrant during the Watergate investigation on Nixon (I can't remember what for, whether it was to compel testimony or provide evidence or what). He went to the White House, was brought in by the USSS who had been briefed that he was coming. They took him to John Dean, who told him Nixon was in a meeting and that he could serve him with the warrant when he got out. Then Dean walked off and they left him standing there in the hallway outside a meeting room, with just a couple of USSS agents standing guard outside. Nixon came out, surrounded by a bunch of other people, and my friend walked up to him, identified himself, amd said he had to speak to him about a warrant. Nixon grabbed him by the the lapel of his jacket and shoved him into an adjacent small room. Nixon was obviously pissed off but was still polite to the agent, and Nixon said he should do something like that when the President wasn't around a group of staffers. My friend served him with the warrant, Nixon accepted it, asked him where he was from, where he went to college, shook his hand (kind of weird), then hurried off.