Ugh, that's the worst. Honestly, I know you don't want to be rude, but sometimes it's not really avoidable unless you want to expend like, stupid amounts of energy tiptoeing around the feelings of people who would throw you under the bus for their malignant spouse, anyways. I can only say what I do, which is that I try to explain to these women that my relationship with my dad is complicated and honestly, it's between us -- not her, me, and him. Because truthfully, it's kind of rude to involve yourself in the intricacies of someone else's relationship with their child, unless you intend to better the life of the child or help the parent right wrongs. Her contacting you for HIS birthday, not even someone else's birthday that you both know, or your own birthday (literally wtf considering I assume she knows you're avoiding him?) doesn't give an indication that that seems to be the case, so eh. I'd feel really weird as a partner to insert myself in someone else's relationship with their child and ever insist their adult child, who has been avoiding them for 2 years, do something -for them-, especially considering this isn't a dire situation or something.
Alternatively, I'd ignore her. I don't really see the point in keeping up communication with people do the bidding for manipulative narcissists.
If it's a punishment gift, he will likely demonize you anyways and make you look bad to her (obviously leaving out the context of your relationship while weaving his narrative as to why you're a terrible daughter, to her), and if you send him something nice (even a card) he'll likely feel validated/use it as an excuse to believe you think highly of him, probably partially undoing the progress you've made over the past 2 years. Just my 2 cents on how it may go based on my experience.