Well, St. Louis wasn't a two-team city under the OP rules since they said 1901. And New York was only a 2-team city.
See, the Browns of St. Louis were originally the Brewers of MIlwaukee because while Ban Johnson wanted a Western city, he sort of needed a stopgap. So, we can give the person who moved them some problems that force them to stay in Milwaukee. And, while Johnson wanted a team to compete with New York, the Yankees were originally the Baltimore Orioles. And, they had a decent chance to stay there since it was a very popular city and Johnson's A.L. would have a monopoly on the D.C./Baltimore area. They just need to convince one of the Giants or DOdgers to jump to the A.L..
Can it be done? It would be a drastic shift but the baseball wars were so bad between the leagues that it is possible that an arrangement would be made wherein the A.L. promises to stay ut of St. Louis (and thus cede the West and South) in return for either the Giants or Brooklyn (which changed their name quite a bit in that era) moving to the N.L..
Now, the Giants were not that well off till John McGraw and his best players jumped ship from Baltimore in 1902. if the Giants are really struggling, it's possible they could move to the A.L.. Brooklyn is also possible, some of that depends on the ownership.
So, that gets rid of the biggest issues - St. Louis was going to be declining pretty fast, and New York needed an A.L. team so they could stay there and switch leagues. Philadelphia, if the Macks hadn't been holding out for so much money, might have kept the Athletics. I think Philadelphia was capable of being a two-team city due to its size, even if one wound up in a suburb or just called that but playing across the river. And, realistically, I think the Dodgers could have been kept there if Shea gets them insted of O'Malley, the Giants are the bigger problem, maybe the Dodgers move to Flushing and the Giants to a suburb, like the football teams moving to the Meadowlands.
Boston is the biggest hurdle, but that's not a problem till the '50s.Someone else can work on those details.