First time I am hearing of kitty. As a heavy tmux user who has had to yak shave a few times due to 256 color support issues, I'm going to check this out!
If you use gpakosz tmux config you may find the following set up useful. https://github.com/lukesmurray/bootstrap/tree/master/.config... I replicated most of the key bindings so you can use splits without tmux. Still no support for multiple sessions or persistence but that’s on kittys roadmap