I can sympathize with why your husband may find it really hard to consider sending your kid to a state school. If it's the norm in his family, for him and his siblings and now all the cousins, it can be really hard to break away from that.
There can be a sense in some families that this is what good parents are meant to do: give up everything to make sure their children have 'the best possible education'. If this is the norm in your family the way of showing love and support for your child it's very hard to break that down.
Coming to realize that, actually, it might not be necessary, that a free state education may have been just fine (and potentially better) for him and now for his child, can feel like heresy. He might feel a huge amount of guilt for not doing it, or not being financially in a position to pay himself.
So yes: I agree with other posters that it would be insane to accept your MILs offer of help, which will come with many many strings attached and cause all sorts of misery.
But in talking to your husband about it, be aware his resistance may not be him simply being a snob, but rather all sorts of deep seated feelings of guilt and responsibility and intergenerational debt. It will be hard for him to go against the grain and (in the eyes of his siblings and parents) deny his child 'the best'.*
*even though it is not actually the case that private = better.