Yes and no.
If the administration says, "For the good of all, we need to get everyone vaccinated; can we do that?" And the answer is "No." Then the next logical question is, "Well, what *can* we do to get the most people vaccinated?" And the answer is "Mandates for Federal employees, OSHA regs requiring vaccination OR testing."
That is not a bad faith intent that the courts will strike down. And that's because the intent to get everyone vaccinated is not a violation of people's rights (as already determined by SCOTUS precedent).
If the administration wanted to, say, get only people of color vaccinated; then that racist intent would certainly not be legal constitutionally. And then, in this scenario, the administration tried to get around that by targeted vaccine mandates for people on medicaid or food stamps as a way to target people of color, but they made up some baloney excuse, then *that* would be thrown out by the courts for having an illegal intent, even though theoretically, the government could technically do that apart from the bad faith intent.