解決

$$$ と記述しエスケープする

[Compose file version 3 reference Docker Documentation](https://docs.docker.com/compose/compose-file/#variable-substitution)
export DOCKER_HOST=`/sbin/ip route|awk '/default/ { print $$3 }'`
export DOCKER_HOST=`/sbin/ip route|awk '/default/ { print $3 }'`

事象

command: >
  /bin/sh -c
  "rm -f /var/tmp/pids/server.pid
  && bundle install
  && export DOCKER_HOST=`/sbin/ip route|awk '/default/ { print $3 }'`
  && bundle exec rails s"

を実行すると下記のエラーが発生する

ERROR: Invalid interpolation format for "command" option in service "rails": "/bin/sh -c "rm -f /var/tmp/pids/server.pid && bundle install && export DOCKER_HOST=`/sbin/ip route|awk '/default/ { print $3 }'` && bundle exec rails s"`