They can sue. They would likely lose.

The reason Liberty is distasteful to some is because of their administration's views and actions engaging in politics and lawsuits to further discrimination and enforce their religious views to those outside of their religion. (remember the Moral Majority?) I don't think that any court will side with a religious institution who claims that they have a right to attempt to legislate morality for those outside of their religion and are exempt from any negative reaction or non-inclusion from those not sharing same. That would be the argument that a lawsuit from Liberty for FBS exclusion based on religion would entail. No one is or has argued that Liberty doesn't have the right to practice their religious beliefs or to say what they have to believe internally and within their own religious function.

Liberty has taken zero effort to break with the past deliberately polarizing behavior of their administration. Key words: deliberately polarizing.

The Sun Belt Conference has no private institutions, does not have any religiously affiliated institutions, and should not (in my view) take the risk of being the first to upgrade a private football program from FCS to FBS. These are not minor points and each one of them stands independently as a valid reason to exclude Liberty from the Sun Belt. In other words, Liberty's polarizing views aren't what's keeping them out of FBS anyway.

There are Sun Belt institutions with a very large online presence so I dont think thats a reason to exclude.

In spite of the reasons noted, Liberty IS getting a serious look from the Sun Belt. Less than one year after stating a desire to upgrade. These things take time for everyone else, why should Liberty be any different?

I dont think Liberty would have any chance at bowl bids, television revenue, or exposure as an independent (assuming they could cobble together a schedule). Would they sue to get bowl access and televised as well?

There could be an anti-trust argument that doesn't mention religion against the NCAA that might be more successful but I think it would be unsuccessful as well.

If they're not admitted to FBS within the first three years of trying because better candidates are available that doesn't mean they've been unfairly targeted or discriminated against either.