There seems to be no source code available for the laby mod which makes debugging conflicts with it a non-trivial task.
What the ReplayMod does to get into the Main Menu is, while not generally encouraged (otherwise the Main Menu will be full in no time), rather standard practice when it comes to changing vanilla GUIs. Whatever labymod's doing, will probably also cause issues with other mods and is afaict in no way the fault of the ReplayMod, so I'd suggest you bring this up to them instead.
When you do, make sure to mention this link to the piece of code that they're breaking, it'll probably be of use to them: https://github.com/ReplayMod/ReplayMod/blob/f36ebf1d0a31fd2591291284dfd35fd7b81e0cce/src/main/java/com/replaymod/replay/handler/GuiHandler.java#L99-L111