It might be useful to try and figure out what's logging the messages.
However, if it was me, I'd strongly consider just starting from your shell in the tty, then running tmux, then starting sway, then attaching to tmux from a terminal emulator.
Thanks for your reply! I've thought about that as well. Haven't tried it though. Two thoughts about it:
1. Running graphical from within tmux feels unsafe (?). Introducing another layer can't be the way to go. BUT this comes from a position of limited knowledge, so I might stand corrected on this one. Also, doing it once for debugging won't do any harm.
2. I'm pretty sure the errors are not printed by Sway itself, but some other OS module. Errors that Sway cause for other modules won't be included in the Sway log. So the problem remains, no?
However, if it was me, I'd strongly consider just starting from your shell in the tty, then running tmux, then starting sway, then attaching to tmux from a terminal emulator.