It turns out that the 1882 Atlantic club did at least flirt with the League Alliance. On January 28, the New York Clipper published the following piece:
So the main difficulty with the Atlantics joining the Alliance was the Metropolitans’ territorial privilege. But on April 1, the Clipper printed rosters for the National League and League Alliance clubs (Philadelphia and Metropolitan). The Atlantics don’t appear—but only because the team’s final lineup hadn’t been settled. They are named as the “third League Alliance club”:
A week later, however, the Clipper quotes Barnie as having applied for membership in the Alliance. The same piece says that the Atlantics had settled on the old Union Grounds; I don’t know whether this would have been outside the Metropolitans’ territory or not.
I’ve checked the Clipper through June 3, and so far haven’t found a definitive statement of the Atlantics’ Alliance membership (or lack thereof). In fact, the Alliance as such was hardly mentioned in accounts of the Philadelphias and Metropolitans; in large part, this is because it was more a loose working agreement respecting territories and contracts than a “league” as we now understand it. Whereas the Clipper regularly printed standings for the “League championship” and “American championship,” I have yet to see any mention of an “Alliance championship” in 1882.