Can you step out from the situation slightly and not be so emotionally involved? Your mil sounds difficult and your DH has conflicted feelings about her. I think he needs your support because continued confrontations will make his life more difficult both now and after she has gone. Your DH probably wants to find a calm way of dealing with her without flip flopping between shouting at her and then embracing her. Right now your involvement appears to make it worse as you appear to want him to chose between her and you. That's horrible for him.
Likewise your DCs will benefit from (careful) contact with her and will probably suffer if they do not have a relationship with her. Is she ignoring your DS because she feels miffed that she couldn't see him after the birth? It sounds like it. Could it be because he is still young, BFing, not so easy to play with? Maybe you have to give it time. Try to encourage the relationship. Dress him in an I Love Granny baby grow or something like that and see what happens.
Because you live 200 miles away, you can control access much more easily. Don't kow tow to her unreasonable demands, do it on your own terms. You can proactively go there for visits and if things become difficult you can leave. You can just shepherd the kids out of the house for a walk to the local park or something and let your DH calm things down. Or get into the car and leave if calming down is not possible.
Your mil seems to feel she had a right to pass judgement on how you bring up the kids which is obviously wrong. I can understand why this irritates you but try not to let it get to you or start a row. Can you find non confrontational ways of doing this. If she starts talking about bottle feeding just smile and say nothing. Leave the room, read a book to your DD, change your DS's nappy, go off and play with the children. Attending to young children is a perfect way of avoiding mean criticising adults!
Finally, stay out of the arguments. Let her be the one who causes trouble rather than you and it will probably get easier. Fortunately she is not on your doorstep, so her interference can be limited.
Good luck.