I work at a university, so I'm going through the same process. They haven't asked about vaccination yet, and also don't provide a resource anywhere that I can find to ask for an exemption (making it extremely difficult to go about this without "outing" yourself). I've got a ton of exemption-backing info and reasons to avoid the jab, but the tests are another matter entirely.

All I can really point to right now is that anything EUA cannot be mandated, and that's what these tests fall under. Every single one of them. And under OSHA 1970 Section 11(c), you may be protected from retaliation if you suspect serious harm could come to you for undergoing the medical treatment. There are reports of people injured after the nasal swab "tests", so maybe try that? Universities are notoriously bull-headed about this stuff, however, and might not care at all. Regardless of liability.