I think that's thin. This really isn't university policy, it's conditions of employment set by the state. WSU didn't have a choice on implementing it, although the details of how to implement were left to them.
The constitutional arguments can only succeed if it reaches SCOTUS. The precedent from that level says that the state does have the authority to implement vaccine mandates with proper justification, which is a test that they'll easily pass.
The argument that there should be a testing argument might have some legs, but I'm not sure many courts are going to weigh in on that, or that there's a legal basis saying it has to be available. And, I'm not sure that's really an issue that can be part of this case. Asking for that here is kind of mixing issues - he'd need the court to rule first that there should be a testing alternative - which is a challenge that should be aimed at the state, not at WSU (WSU didn't create the mandate). Once they make that ruling, he can claim that he was unfairly damaged by WSU...but WSU's immediate defense is still that they didn't create the mandate, and if they'd been allowed a testing option, they would have offered it. If the court did not rule that there should have been a testing option, the unfair damage argument immediately falls apart, because he didn't meet the mandate.
Of course, everything changes if there really were some sort of shenanigans. But, the requirements were very clearly communicated and he didn't meet them, so proving any sort of discrimination is going to be a very tall order.