Please don't assume that the bride is expecting you to give her a gift.
She has reached out to a family member to include them in her wedding. That's all.
The idea behind "you have to buy a gift if you're invited to a wedding" is this:
In nearly every culture in the world, weddings are a Big Deal in Someone's Life.
Big Deals in the lives of people {you} -care- about are marked with gifts; weddings more so than anything else. (think about it--wedding gifts are cross cultural)
If {you} receive an invitation to someone's wedding, they are telling {you} they care about {you}. It is then presumed that {you} in turn care about them. (Might not be accurate, but it's not an unreasonable thing to hope for.)
Put these together, and it's then society's expectation that {you} will mark the occasion with a gift in order to tangibly indicate {your} affection and good wishes.
The gift is the "word" that the culture's "dictionary" uses to say, "I love you and I wish you well on this Important Occasion." It's the "Lego block" of the society's etiquette/cultural language.
It's not the couple, or their parents, who are creating the expectation of the gift; it's society in general.
If {you} reject this expectation, there are of course other ways to indicate {your} affection and good wishes.
Whatever you do in the face of this expectation is then a piece of communication about how {you} feel about the person having the Big Deal happen in their Life. It will carry a message.
If {you} do attend the ceremony, that's a message that these people and this Big Event are important to {you}, and so for you to -not- send that "word" from the "dictionary" is hurtful. (That's where the "you only 'have' to send a gift if you attend the ceremony" comes from, and that's why society's pressure is so high.)
If you attend their reception, and eat and drink their food (and presumably celebrate as well), and you reject the culture's accepted form of communication (giving a gift), you send the message that you will take (their hospitality and their emotional connection), but you won't give anything (a gift, or an equivalently powerful message of regard and good will).
Those are hurtful message to send. And of course the person receiving that message is the couple (and any family members), not society at large. That's why a couple (or family members) may be offended and think less of you.
So in this case, think about what message you want to send on the occasion of this Big Deal in the life of someone who is indicating--14 years after the family breakup--that they do consider themselves to be connected to you in a positive way. How do you feel about that? What would you like to say? Would you like to reinstate some sort of relationship?
In your case, you don't want to use that particular "word" from the societal "dictionary." So you'll need to find some sort of "terminology" in your own "language" that will send as powerful a message.
If you DON'T want to have any relationship, then choose a communication that will create the effect you want.
Also remember that perhaps you don't believe in weddings--in our culture, we have a very strong presumption that most of our beliefs are specific to us, and that it is not appropriate to demand that other people follow our beliefs.
So Christians attend weddings or bris/baptism/dedication-type events in synagogues or mosques or all sorts of other places (vice versa all over the place)
in respect for the beliefs of the people they care about. They send messages of good will to people who are publicly declaring their belief in a diety they themselves reject.
Because in our culture, beliefs are considered to be a matter of one's OWN conscience.
So if you ignore this wedding, or if you mention how you don't believe in weddings, that's going to be incredibly hurtful "language."