Simplesmente pule a carne para o tee. Ou eu entendi mal o problema?
root@el6 ~ # cat test.sh
date
root@el6 ~ # cat test.sh | bash | tee >(logger -t "test")
Wed Jul 10 23:08:03 NZST 2013
root@el6 ~ # tail -n3 /var/log/messages
Jul 10 23:08:03 el6 test: Wed Jul 10 23:08:03 NZST 2013