I had read your post earlier and was thinking about how to reply when I come back and find that Wings has done a great job, along with Uny, in replying to you.
If you are willing to turn your children over to the State to be indoctrinated by them, then you have little recourse to avoid complying with their "rules".
I think that the doctor's visits are really the least of your problems in this situation. The damage that will be done to your precious children in that environment is of much greater concern.
Children are not able to discern what is best for them and to act on it. That is why they have us to protect, nurture, and train them in the way they should go. My children may "like" to eat candy all day or watch videos for hours on end or do any other number of things that I know is not good for them. It is my responsibility as their mother to provide structure, boundaries, and protection. This is especially important while they are young. They do not yet have the moral foundation on which to stand against the evil in this world. So I must make decisions for them, even when they may not "like" it.
And Wings is correct in that being in a room full of children the same age is not "real life". Real life comes from living and working as a family in the home- planting a garden, playing in the pool, making dinner, helping to change the baby's diaper and put her down for her nap. Learning to relate with people of all different ages and backgrounds. Learning to work, to serve and to get along with others.
So I would just encourage you to take a good hard look at the situation and do what is best for your children. It is obvious that they are important to you and you want the best for them. If you really want to protect their rights and yours, you must not turn them over to the State. You said you could homeschool, so do it! Thank God you still have that freedom and take advantage of it!
There are many resources available and places to find support and social time for your children when you homeschool. And then you are able to decide when, where, and how much they will be exposed to. Even in the small town where I live we have started a Coop and had over 50 children attend our last session!
And please know this is coming from someone that is in the thick of it so to speak. I have 6 children 6 months to 10 years old. They are with me 24/7. I am right there fighting the good fight as well! I know exactly what is involved in what I am encouraging you to do. For it is what I spend my life doing. My days are full! But I refuse to let the government have any more control in the lives of my family and children than absolutely necessary.