Are you using Bash or another shell? There's nothing different about wsltty to cause it to not load that file if you're using Bash.
It works great with Docker Compose. It's what I use personally and I don't manually source any files. I just open wsltty and start running Docker commands (or any Linux command).
There is a config file for mintty but nothing is needed to be configured to get your PATH set up. I have my wsltty config in my dotfiles at: github.com/nickjj/dotfiles/blob/ma...
It's mainly just settings for various wsltty specific UI things.
All of the PATH related things would be done inside of WSL (unrelated to your terminal).
Maybe double check your ~/.bashrc file to see if the PATH is being overridden there.
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