I know this is an old question - but in case you did not get any answer....

I just started using Byobu with PuTTY, and found that I can use Ctrl+A, % to split the screen vertically and Ctrl+A, | to split the active pane horizontally.

These key commands are based on the screen program. You can find more at the screen man page.

Obviously Ctrl+A is the prefix I'm using, change that to whatever prefix you are using.

Hope this helps someone!