While it's not my area of law, and it's possible that I missed a new opinion, AFAIK federal funding has never been pushed this far. It would allow the central government to get too involved the affairs of institutions.
For example, a court can delve into the sincerity of a person's religious beliefs, but not the truth thereof, or whether a doctrine is heretical to its institution.
If the federal money were highly directed rather than general, or targeted to another area, such standards *could* apply. For example, if the feds funded a lecture series, they could require neutrality or breadth in the series (and probably could not forbid it), but general student aid, or a chemistry lab, won't allow intrusion into political or religious affairs.
There *have* been programs (unconstitutional, IM!HO) broadly denying categories of funding, such as denial of all or parts of highway funds for speed limits or DUI laws, or Title IX (which has been badly abused), but these are far cry from a private right of action under the First Amendment. So far, in fact, that I would expect Rule 11 Sanctions to be more likely than success.
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