TL;DR You can now include a wppusher.php file in your WordPress packages with all the shell commands you wish to run after WP Pusher has installed or updated a plugin or theme. That includes running Composer. Also, WP Pusher now supports GitLab!
Composer seems to be something everyone is talking about in the WordPress community these days. It is a great tool for developer productivity and code reuse. However, if you are trying to use Composer to distribute your WordPress product to end users, you might be doing it wrong ™.