Exactly what outcome do you expect to take place if you were to win your lawsuit? I mean, as opposed to what might happen if the vaccine mandate was successful and a significant number of people were vaccinated who wouldn't have been otherwise, what would your lawsuit accomplish if it were upheld and these same individuals who would have been vaccinated weren't? What would this accomplish? What aspect of the general welfare of the nation would be enhanced if you were to prevail in your efforts?
...if the vaccine mandate was successful? That isn't going to happen so there is no point in speculating about what might or might not happen under those conditions. The Governors of these states are vaccinated and are pro vax. They are opposed to "forcing the vax" on anyone because they know it won't work. A law requiring the vax might work if it were passed by the house and the senate and signed into laqw by the pres, but a Mandate is not a law. It is not supported by the constitution and it will not be supported by more than half the folks in the country.
Another problem with mandates, other than not having the force of law behind them, is that when you is that a change in administration can result in the cancellation of mandates. Congress has gotten lazy and can do nothing permanent. Mandates are like a christmas tree. Pretty to look at, but nothing on them is permanent.

Liberals are like a "Slinky". Totally useless, but somehow ya can't help but smile when you see one tumble down a flight of stairs!