You shouldn't have to rethink that position.
There's a difference between speech/belief and BEHAVIOR.
Laws and courts already make that distinction.
In order to have an open, free, and just society, we need to protect the right to speak openly, even if those ideas are considered stupid by other people. The point is not to protect stupid positions of individuals, the point is to preserve openness of society. Ideas being "dangerous" is not sufficient justification to pay the heavy cost of being a closed society, especially when better education is an alternative.
But in terms of behavior, this is a good case for law being a solution. True libertarians (i.e. not anarchists in disguise using the "libertarian" label to avoid being dismissed immediately) believe in Occam's Razor of government intervention: that laws do have a purpose, but government should get involved only when there is genuine need, and in the degree and proportion necessary to solve the problem, where there is the problem, and only if government is actually capable of doing so rather than finding excuses to soak up tax payer money and infringe on liberties despite knowing they can't solve the problem.
In the case of diseases and epidemics, it is no longer an issue of personal choice and responsibility of an individual, because literally a single irresponsible choice can lead to the deaths of millions who were not given the opportunity to make a choice for themselves, and the damage can continue INDIRECTLY. This is quite unlike any other risk-factor people are usually afraid of (even guns, where attackers must be in the direct vicinity of the victims, and there are ways to increase security and defense).
So the solution is that in terms of beliefs and speech, yes you are free to openly, strongly disagree with vaccinations. But in terms of behavior, you and your kids are getting vaccinated anyway. You can disagree all you want while you are getting your mandated vaccination.