I was really excited to read this new development. Not sure if I'm reading this correctly, but your H's decision to include you with D and his parents on his birthday feels 'intimate' to me.
I agree with CW. You definitely need to bring a birthday card! You can't go wrong doing what you would normally do for a friend. You could sign your name with D's name...or....sign only your name. Let your instincts guide you, but this could be a safe opportunity to try to move toward more intimacy with H in a non-threatening way. Jody always said that humor was a good way to bond with my H/XH. She said that laughter causes the release of oxytocin, the 'bonding hormone'. This neurochemical is released right after orgasm and in mothers during nursing. I would suggest a humorous card. That's non-threatening.
I also think that a sweet, personal, non-threatening present would speak volumes to H in a way that won't send him running the other way. The added bonus is that when he looks at it (back home in his 'bat cave') he will think about you.....NOT OW. I'm thinking that a home made gift certificate from you and D..........or you and the cat/dog.........something that's non-threatening to H..........might work. I gave H home made gift certificates for (1) a movie and (2) for me to look in on his mother for his first birthday after the bomb. Maybe you could give H a gift certificate for a home made dinner prepared by you and D, or a movie, or walking his dog....or something like that. This would give him the opportunity to initiate a social outing with you if he chooses.
My 2 cents.
Please let us know what happens, OK? We are pulling for you!
