There doesn't seem to be much on this, but SecureCRT can work with Byobu/tmux/screen. The issue I encountered with it was that the screen kept raising the current line every second or so. Something regarding scrolling.
Anyways, the fix for this is under Terminal>Emulation>Modes in the Session Options/Properties. Under Initial modes, delect everything in this list. I also set the Terminal Emulation (Under Terminal>Emulation) to Linux with ANSI Color, but I doubt this has anything to do with the line feed issue.