Falls Road Running has traditionally tried to enter competitive teams in the RRCA Club Challenge Ten-Miler in February and the Rockville Twilghter in July. From time to time FRR will sponsor competitive racing teams in the Baltimore Running Festival.
In my mind there is no such thing as a "non-competitive" team. There may be teams that realize their limitations against more talented runners, but once you join a team you are generally in the event to excel at some goal.
Feel free to email firstname.lastname@example.org for any questions.