I don't think you "should" necessarily. You can if you want to. Hopefully it does not impact the musical result. You do, after all, need to trade voices between hands when playing some of the Bach fugues.
I typically play this sort of stuff as written unless I feel there is a good reason not to (if it's uncomfortable, for example). I like using physical gestures that follow the musical contents if I can, it just keeps me more in the music somehow. In this case, the "crossing" chords belong to one layer of the music with its corresponding voice leading and I like to consistently keep it in the same hand for that reason. Likewise, it keeps the left hand "um-pah" is another layer consistently played by one hand.
But I know some other pianists don't care for this approach and just distributes stuff as they feel is convenient.