Can the order of Byobu tabs be rearranged?

Only with tmux backend:

Ctrl + Shift + F3 move window left

Ctrl + Shift + F4 move window right

You can only move one step at a time but the process is quite intuitive and fast.

While Byobu doesn't have a hotkey for this, it's possible to move windows using the underlying Tmux functionality. See the manpage:

move-window [-ardk] [-s src-window] [-t dst-window] (alias: movew) This is similar to link-window, except the window at src-window is moved to dst-window. With -r, all windows in the session are renumbered in sequential order, respecting the base-index option.

Full disclosure: I'm the author and maintainer of Byobu.