The younger kids are her siblings too. She feels they are family. They are going to be important to one another when you and your husband are older and not able to do so much, and not on the same wavelength. They will be of the same generation. Five years will be nothing as an age gap when they are 35 and having kids at around the same time - probably. Probably, the younger two would like her around, and the elder sibling too; your DH wants her around; it's just you who wishes that you could have a family which isn't really blended.
The thing is, that if the family relationship isn't fostered, then this child may find she belongs properly nowhere. People will just be waiting for her to move on, get a job, get a life. This is harsh. But we live in quite harsh times, I accept. When things are tough, families do perhaps want the teenagers to grow up fast, think about work, get independent. This is a bit, though not exactly, obviously, like living in 1920s Britain, when girls probably felt they ought to stand on their own feet.
Maybe DSD mum is having a hard time with money or something, and life for DSD at home is real and gritty, and as it is, but a bit bleak. Maybe her mum is a bit down and always going on about the harsh practicalities of life, and dreading how she is going to find more stimulation and development for her daughter. Perhaps her dad's home - and yours of course - seems more fun and secure.
Anyway, she is with her mum all week so why shouldn't your DH host her every weekend? Why shouldn't he do 50/50, actually?
Do you, unconsciously, not want to take DSD out with your two, as it will mean spending more money on cold drinks, ice cream, pizza? Why wouldn't it be more fun for the younger two to have their sister with them on outings?
Yes, DH should interact a lot with her, and his friends could too, when they are round. They could easily happen - the different generations hanging out together.
I suppose, the optimum is it just being her home, and her not having to feel like a visitor.