tmux: how to bind a key to launch shell command?

The right answer to this question is: use bind-key, e.g.:

tmux bind-key "$KEY" run-shell "/path/to/script.sh"

where KEY=C in your case.

C-b c already has a standard binding which it might be wise to leave unchanged. Choosing another character, eg C-b C you can setup a binding in your ~/.tmux.conf file as follows:

bind C send-keys -t.- 'mvn install' Enter

The -t.- means "the other pane". Enter stands for the key of that name, i.e. the newline at the end of the command.