Not sure it's really necessary to feel you alienate New Brunswick or PEI CFL fans if you don't regionalize the name . IF your that fickle about seeing a game I doubt you were going to patronize the league anyways .
If it's Private only money they will call it whatever the like .
I have been told by more than one person who has lived in the Maritimes that their sense of geography is very constricted compared to pretty much the rest of Canada. While people like myself would think nothing of driving 4.5 hours to Regina for a 3 hour football game, then driving 4.5 hours back to sleep in my own bed that night, to a Maritimer, a trip half that distance is considered insanely long, requiring an overnight stay.
I was once in a community theatre production of a Canadian play, written by a Maritime author, originally set in the Maritimes. We changed the setting to Saskatchewan, using cities and towns that were approximately the same distance apart as the ones named in the original script. At one point in the script, my character's mother reveals plans to move to another city, and my character complained, "But that's so far away, mom!" In the script, it's intended as a dramatic moment. The line drew laughs from people in the audience, because in Saskatchewan terms, it was a hop, skip and a jump away, maybe 2 hours drive.
For the Atlantic team to work, it must draw at least occasional support from fans up to 5 hours drive away, much like the Roughriders. If you name the team after the host city, the way a Maritimer's mind works, people in Dartmouth won't support the team, never mind Moncton or Charlottetown, because that's not "their" team. Therefore, to increase the likelihood of regional support, it must be a regional name, either "Atlantic" or "Maritime."