It's well documented that the St. Louis Browns researched thoroughly a move to Los Angeles: owner Don Barnes and others looked at the logistics during the 1941 season and had their ducks in a row to put it to the American League at the winter meetings in December 1941. Of course, Pearl Harbor put an end to that.
But suppose the idea had been floated during the abysmal 1937 or 1938 season, and Barnes had done due diligence then? Would the AL have gone along with the idea? I would bet the other owners would, if only to get larger crowds and thus more revenue from the erstwhile Browns. Moreover, the notion of traveling to Los Angeles (= Hollywood) three times a season (instead of four, given the constraints of rail travel) would likely have been a selling point: imagine, for example, Clark Griffith rubbing elbows with W. C. Fields or Groucho Marx.
If there's any interest in this, a timeline could result: could be that there might even be a World Series on the Left Coast in the '40s.