> If the script is running the command, is it somehow running it through a terminal?
No, it's going through whatever shell you specify in the shebang on the top line. You should probably detach the process from the startup script. For bash, that would be adding a `&` to the end of the command. Otherwise that process will be hung until you exit out of kitty window. That's also assuming that kitty won't automatically create a detached process when it's run directly like that.
I believe there's a way to auto start certain applications in specific windows, but the documentation site doesn't appear to be working right now, so I'm not sure.