From your description, you're using byobu with the GNU Screen backend, where "splits" are not really handled very well.

Byobu's tmux backend provides a much better experience when trying splits.

You can switch your backend by running:

byobu-select-backend

Or you can manually launch with either backend, using byobu-tmux or byobu-screen.

Full disclosure: I am the author and maintainer of Byobu.