This is all subjective though isn’t it?
People saying the daughter must be feeling x or y or being made to feel a certain way by the dad etc when actually by the OP insisting on a different approach the daughter will be pressured into feeling something by her mum. There are no winners here except perhaps the daughter.
And if adults start referring to their new partner (bf/gf) kids as their step kids and don’t assume it’s too soon why should the children be made to feel that it is?
And people saying that nine months is too soon is all subjective as well. Yes it might be, but equally maybe the relationship will last and the bond between the siblings will strengthen. At thirteen the DD will know that her parents potentially have new relationships. She may actually want to be included in what he’s doing because that makes her a part of his life. This isn’t like when you have little kids and you hold off forever for the kids to meet the new partner. Older kids are more knowledgeable than that.
And every relationship is different. My DC was ten when I got together with my now DP. I introduced them weeks into the relationship not because I wanted to, but because my eXH gave me an ultimatum, either I tell DS, or he would. So I made very gentle introductions, and nearly eight years on they still have a very good relationship.
By the same token though my eXH did the same and introduced his GF and her DC to him within weeks, moved her in and had another child with her within a year and DS feels totally differently about her, in fact he has absolutely nothing to do with her and refers to his dad’s DC as his step sibling not even half sibling.
He also refers to the GF’s DC as his step siblings for ease of reference rather than anything else. He has literally nothing to do with them, ever. He does see his half sibling with his dad sometimes, but even that is rare.