You do not need to include Composer in your PHP image. Composer can run as a standalone (temporary) container to do any actions you wish.
Installing
Development
docker run --rm --interactive --tty --volume ${PWD}:/app composer install --ignore-platform-reqs --no-scripts
Production
docker run --rm --interactive --tty --volume ${PWD}:/app composer install --ignore-platform-reqs --no-scripts --no-dev
When running in two separate containers it was important to make sure that the NGINX root path and the PHP-FPM public/ path were the same.