As a half-sibling whose mother was often treated really poorly by adult half-siblings, and myself excluded once as a young teen by some of my half siblings, I can tell you it was incredibly hurtful; and I never forgot/forgave any of it. How do you justify hurting a child as an adult? It’s mean, and ugly.
You all need to stand as a unit, including DH, irrespective of whether or not he’s paying for the wedding. TBH I’d leave my husband over this, because you’re a family of 5, not 4. Your SD’s pettiness is mean, and divisive and she should know better!
I would not let any of my children attend the wedding, and would die on this hill, and go so far as to leave DH if he let it slide. You DON’T exclude a child, because they aren’t a “blood” relative. The fact that she isn’t even paying, makes it even nastier.
How exactly would you explain this scenario to all 3 kids? Sorry, you can go because SD likes you, but doesn’t like you, nor mummy and we aren’t considered family - that is ultimately what it comes down to, and how it will be perceived.
Your SD has chosen to put her father and his family in a difficult position on purpose. He needs to withdraw both himself, and the money for the wedding. Family is family.
She’s an adult and can’t even respect her father married who he wanted to, but is marrying who she wants to, and expects to be treated like an adult with respect to her decision.