The first thing to note is that a vaccine mandate, even if ill-advised in some cases, is not per se or intrinsically immoral. Most Catholics acknowledge this in the case of other vaccines. For example, few complain about the fact that schools have long required measles, mumps, and rubella vaccines as a condition for attendance. Whether or not it is a good idea for a school, a government, a business corporation, or any other authority to impose some particular vaccine mandate is a matter of prudential judgment. Hence, the Covid-19 vaccine mandates cannot reasonably be objected to simply on the grounds that they are mandates. A reasonable objection would have to be based instead on the judgment that they involve a failure of prudence.
But how prudent or imprudent a policy is is a matter of degree. A certain tax policy, for example, might be extremely wise, merely defensible, merely ill-advised, outright foolish, or extremely foolish. The same thing can be true of a vaccine mandate. In my opinion, Covid-19 vaccine mandates of the kind now in play in California are somewhere between ill-advised and foolish. For one thing, I do not believe it has been shown that such mandates (as opposed to voluntary compliance) are necessary in order effectively to deal with the virus. That suffices to make them a bad idea, because imposing a vaccine mandate is a significant enough infringement on personal liberty that the authority imposing it faces a high burden of proof.
For another thing, when citizens are highly polarized about some policy that has merely prudential considerations in its favor, that is itself a serious reason for a public authority not to impose it, especially if the skeptical part of the population is already distrustful of the authority and sees the policy (whether correctly or not) as a crisis of conscience. This is just basic statesmanship. When polarization and distrust are already very high, the aim should be to reduce them, and to try as far as possible to accommodate those who have reservations. Heavy-handed policies like vaccine mandates will inevitably have the opposite effect.