Having read through the tutorial on the documentation again, I realized the key is to create a new level for the main menu. So this problem got solved.
Another issue, probably a bug of the editor, is that the per-level game mode does not seem to work properly. Whenever I change the FirstPersonGameMode’s default pawn class to character, the default setting also changes. My workaround is to create a duplicate of FirstPersonGameMode, name it differently and set its pawn class to character for the main menu level only.