False line of thinking. To compete at the FBS/BCS level of football, a university must qualify as a division 1-A institution with the NCAA. To be divison 1-A, you have to carry a minimum number of both mens and women's varsity sports.
An athletic conference consisting of only Big East parochial basketball schools would leave Notre Dame very much short of an adequate athletic conference to maintain division 1-A status and NOT have to go through the mess of sheduling multiple varsity sports entire seasons as independants.
The extent to which this is actually a problem is overstated. They have golf, rowing, tennis, track and field, swimming and diving, and cross country which are already scheduled "like an independent", because that's how everyone schedules in those sports. There would be no Big East competition in ice hockey or fencing because of the low participation rates of those sports.
That only leaves basketball, baseball/softball, lacrosse, soccer, and volleyball that Notre Dame would have to worry about. If Notre Dame were to stay with the Catholics, then men's lacrosse is set, because of the 8 teams that will play Big East lacrosse, only Syracuse and Rutgers aren't in that group (Villanova, St. John's, Georgetown, Providence, Marquette). 7 of the 8 Catholics play softball, and all of them play men's and women's soccer and volleyball.
That just leaves baseball and women's lacrosse. The current WLAX Big East is 4 "football" (Louisville, Rutgers, Cincinnati, UConn), 4 Catholic (Georgetown, Loyola, Villanova, Notre Dame) and Syracuse with their foot out the door. However, WLAX has at least two lacrosse-only conferences (the NLC, comprised of mostly recent upgrades to the sport, and the American Lacrosse Conference, which has six major schools plus Johns Hopkins, who is a major when it comes to lax), so Notre Dame should have no trouble finding a hold in either. As far as baseball goes, there's only four of the Catholics that play including the Irish, so...invite Xavier and Dayton for all-sports, and call it a day.