I live with my dh in a house he bought with his ex girlfriend. Although they only lived together here for a year, and we've now been together for 11 years, it still doesn't feel like my house.
We've done loads of work to it, re-decorated throughout etc, but to be honest, because it's not a house I would ever have chosen, I don't think it will ever feel like mine.
You have extra layers of complexity due to the relationships of his children and the house, so I would think really, really carefully before agreeing.
Given my time again, I would have insisted that we choose a home together and sell this one, even though it made no sense financially.
I think that you need a new start in your new realtionship in a home that has been chosen by both of you.