There's the rename-exe option in mise.toml to rename the docker compose binary to `docker-compose`. I've used it on some machine, I think. Look it up in the docs, it doesn't take long to wire up.
The default docker-compose definition comes from aqua:docker/compose though, which does not support rename_exe. You have to specify github:docker/compose instead.
rename_exe doesn’t do anything on the aqua backend either (and isn’t a documented option). I’m on mise 2026.5.10 right now though so if you fixed something recently I might be missing out!