A bit of an improvement: do space, V, enter instead of 0, space, $, enter. A single V is a lot easier to type than a 0 and a $. V is equivalent to vim's "do visual mode select of entire line" command.
I can't speak to byobu directly but since it's just fronting tmux, you could also bind a key to all this in native tmux. Something like this:
bind-key -r -T copy-mode-vi Y send-keys -X begin-selection \; send-keys -X select-line \; send-keys -X copy-selection Then you just enter copy mode, go to the line you want to copy, press Y then Enter. The line will now be in the paste buffer. (While that's a bit long and ugly it does work for me.)
Edit: Tested with tmux 2.4