So to put my cards on the table here, I spent many years as a research administrator at a major University (no reason it needs to be a secret I guess, it was the University of Chicago). In that capacity I helped departments and faculty members through the maze of administrative, contractual, legal, and more often than I liked (or was qualified for frankly) financial issues relating to the seeking and administering of research grant funding. The lion's share of my work was with corporate entities and state and local governments, which is a much smaller part of the pie than federal grants, but is growing and presents some unique challenges vs the generally one-size-fits-all nature of federal grant administration. I now do broadly similar work for a non-profit, in case you're wondering.
All of that is to say, the process and methods of acquiring research dollars is a world I know intimately, it's been the majority of my professional career.
Which is prologue to the answer to your question: there isn't one.
At the basic broadest level, when it comes to performing research, Universities are a collection of faculty members who act in a way analogous to independent contractors. That's not strictly legally true, they are employees, but they dictate the terms of their work and the direction of their research and their choice of collaborators with a degree of input from their faculty leadership (departmental deans and chairs and the like) but functionally none from main campus.
And the research belongs to these faculty members. If the professor leaves Illinois and goes to Texas, the grant travels with the faculty member. So there is a very competitive marketplace for rainmaking faculty who have existing big grants and the connections to the federal granting agencies (and others) to continue to bring in more.
I don't know how else to say it other than that athletics and conference affiliation has zero to do with this. The Big Ten Academic Alliance (formerly the Committee on Institutional Cooperation) makes for a lovely talking point, but there's no there there. They'll hold conferences between the purchasing departments, the facilities management departments, staff essentially, networking events, share best practices, that sort of thing. I recall that U of C had a license to a certain software program used for screening against internationally sanctioned parties (that stuff does come up, these Universities do projects all over the world) that was a group rate the Big Ten put together with the company. A nice little savings, and perhaps something other conferences don't do (I have no idea), but this is the smallest of possible potatoes and has nothing to do with research.
You don't have to take my word for this, go look at their website.
The stuff they do is incredibly small bore and not meaningful for big-dollar research.
(EDIT: As an aside, there are similar ad-hoc groupings of schools that do similar staff collaborations and the like unrelated to sports. In addition to the Big Ten stuff, U of C was a part of an "Ivy-plus" group, a Chicagoland group, these are all just email listservs that get together in a hotel ballroom every year more or less)
A consortium that structured or forced research collaboration doesn't exist and couldn't exist, in the Big Ten, Ivy League or anywhere else. That's not how the system works, it's antithetical to how the system works.
Anyway, that's my spiel which I felt compelled to give due to the specificity of that question. There are all sorts of interests at play (financial, personal, all sorts) in this conference drama, and it's not as if academics plays no consideration when so many power players are University presidents. Universities are unruly, disorganized entities, they do lots of things for conflicting or unclear reasons. But I don't want people here to think that there are big federal grant dollars at stake in these decisions, that is outside the scope of anything that's happening.
Fundamentally this a battle of athletic departments for TV dollars, and always has been from the day the Supreme Court opened the bidding for TV rights in 1984.