This is another case of “should have read the manual”, but there are ways to see previous output in both screen and byobu.

In screen: ctrl + a + [
In byobu: F7

These key combinations cause screen/byobu to enter copy mode, where you can navigate using vi keys, can set marks for copying using return and the like.

This should have been blindingly obvious, but I guess years ago I got used to using screen without scrollback. There more you know… the more it seems that you don’t know anything at all.