Totally accept what you say OP about your limited choice. It was the same for us a few years ago, and one of the reasons we moved.
I think withdrawing your children will draw more attention to the issue than letting them participate. It's a good way of getting your child to understand that sometimes it's easier just to fit in, especially when it's no skin off their nose! If they don't go to assemblies or have to leave part way through, they are bound to get questions from other kids, so you would need to prepare them for that anyway. Did you realise that non-CofE schools also sing some religious songs, are expected to have some kind of "act of worship", do Harvest Festival and celebrate the festivals of other religions such as Diwali and Eid?
Can't you just explain about religion to them? We are atheists but I always explained to my daughter (since before she started school) that mum and dad don't believe in god, although various other members of the family do. I always answered any questions from my point of view, and others' points of view.
For example, if you don't believe in god you can't believe that christ was the son of god - but you can still understand christianity and other religions as political movements with charismatic leaders who spread an important message that it is wrong to demonise the poor and disadvantaged in society. I have always treated it as something to learn about (rather like crossing the road, or what the Ancient Egyptians believed) and pointed out she would decide herself what she thought when she's older - rather than telling her what she should think. That way, there's less pressure.
She's now 10, understands religion, has learnt about them all (or most of them) and respects the fact that some of her friends believe in god, and actively worship at church. They accept that she doesn't believe in god, or go to church. It's just never been a big issue. She knows the Easter and Christmas stories, which are the most important in the Christian religion. But like us, she lives happily without religion in her life.
Recently a family member was married in a catholic church. She was a bridesmaid, and was blessed as a non-believer while all the catholics were taking communion. These things happen, and there's simply no way you can make a fuss in that situations without embarrassing your child. When I mentioned to her that I was proud of how nicely she dealt with it, all she said was "The priest had really sweaty hands"!
I think it's important to teach tolerance and understanding of these things. There's no way you can avoid religion in our society - better than they learn about it from you. There might be some appropriate books you can get through the Secular Society or something like that - but I never felt I needed them.
Sorry this turned into an essay!