I can't think of any reason why, or even how, Kane and Bernstein would use an intentional fall as a way of helping to seize control of the newspaper.
If you're looking for some meaning of that moment, to me it accentuates the relationship between the three characters of Kane, Leland, and Bernstein. When the trio arrive at the newspaper, Kane and Leland are riding in the front of the stagecoach-thing (?) while Bernstein is on the carriage behind them, sitting on top of the furniture that he later trips up with. Kane and Leland head inside together to confront the newspaper editor while Bernstein is tasked with bringing in all the gear from the carriage.
This puts Kane and Leland on an equal level, especially when the editor shakes Leland's hand first, mistaking him for Kane. Jed Leland is a strong character and one of a few who can stand up to Kane as the latter loses his way. On the other side of the coin, Bernstein is like a lap dog to Kane. Unwaveringly loyal but also always in his shadow. So while Kane and Leland make a dignified entrance to their new newspaper as powerfully idealistic young men, Bernstein--a follower--is less than graceful in his entrance.
I also think it's a little bit of slapstick. Shortly afterwards this is expanded as Kane and the editor are arguing in the doorway of the office while Leland and Bernstein push past the flustered editor. Kane is assertively taking over in more ways than one. He is also an avid hoarder; the amount of belongings he brings in and the way he uses them to assert himself in the newspaper is emphasised comically in these scenes.