Celebgran, you have to follow your heart and also listen to a calm mind when making decisions like these.
You're right about not knowing whether your gifts are given to the children, or what information they will be brought up being told about you. It sounds as though the process of buying and delivering gifts is very painful. I can understand you wanting to avoid it at the present time, but you may feel differently at some future date.
Have you considered opening a bank account , put in whatever money you may had spent on gifts, treats for the children. Write birthday and christmas cards, age appropriate, each year that passes. I wonder if it would be possible to place those with your will, so that if the worst happens, and you aren't able to build a relationship with the children, they will at some stage know that you loved them. I hope this doesn't sound over dramatic, or maudlin, but there will come a time when the children want to know more about you. If you avoid any negative comments about how the relationship with your daughter broke down, I don't see how this could cause harm.
Whatever you decide to do, I hope it helps make you feel less hurt. Rock and hard place is a cliche, because it described situations like this so well.