Sodalitas is probably the one you want.
Fraternitas seems classically to have referred only to the quality (as in "Libery, Fraternity, Equality").
But my medieval times, while it retained that sense, it could (according to Niermeyer) mean a religious community, as of monks or canons, a "community of prayers or good works" or the membership of such a community, a "merchant or craft guild", or "a sworn association in a city [or] commune". So if fraternitas sounds better to you, I don't think it would be incorrect.
You might also consider using just Fratres "brothers", somehow qualified, as in, say Fratres Viciniae Veteris, "Brothers of the Old Neighborhood".