Treelover, as with many OPs, you only gave us half the story in your OP. As you have given more information in later posts, the picture has changed.
It is clear that your DiL, in her relationship with you, is simply playing out the bad relationship she has with her own mother, She is oppositional to you because that is how she relates to mother figures.
She became a vegetarian to annoy her mother. She is an aggressive meat eater and dog hater to annoy you. She simply doesn't know any other way to relate to older women.
The problem is hers not yours. I suspect that the best thing to do is to make the most of the relationship with your DD and for a while do not attempt to have any relationship with your DS and family, that doesn't start with them.
When they do approach you be open and friendly, but stand your ground with them, if they try to emotionally blackmail you, do not submit to any demands from your DiL that you serve meat etc. I think that giving way to her demands, will make the relationship worse rather than better, as she will then hold you in contempt for being weak.
There is no easy solution to this relationship problem, you may never have a close relationship with your DS's children, but your best hope is leave him and his wife to drive the relationship, offer nothing, no advice, do not offer buy things for the children, nothing, but if they ask for help then offer what you can without compromising your principles. If they ask if you would buy the children a jacket, or shoes etc, insist that they are precise about what they want and come with you to choose them. If they want you to feed or buy them ham sandwiches then refuse, because it is against your principles.
Hopefully, if your DiL can see that she cannot intimidate you or bully you, but that you will ignore or refuse without emotion unreasonable demands on you. She may become more amenable.