Hi!
Do I send a card? I know I’m damned if I do, damned if I don’t.
In situations like this, I say do what you feel is right for you to do. Given the fact that you are going to have LC with them because of your mom, I would lean more towards offering a small olive branch (like sending the card) with a fairly neutral birthday wish. Keeping things calm and neutral, using the tools we talk about here will make the LC much more pleasant and easier to deal with.
What do you think?