I don't think there's any good reason to send the card. If you send it without a name and address, not only is she probably upset that you made contact, but she doesn't know for sure if it's you or not.
You are right. It's a fact of life that we have to sometimes sacrifice our feelings for someone else's. It's like in a relationship - when one person really wants to break up and the other doesn't...there's not much you can do.
At this point, your mother knows you have tried to be in contact. If she ever changes her mind, she knows what she can do. I know that's a tough thing to hear.
The important thing is to not look back at this and think that there was something wrong with YOU that caused her to give you up, but you seem relatively well-adjusted (in a perverted sort of way), so I'm sure you're already confident about this. Plus, anybody from Nippleville gets my vote of confidence.