It's not a crazy suggestion at all. I would keep gently offering, possibly her mother will eventually take you up on your offer. But she may not. Not everyone can see past their spite.
My family includes my two stepchildren whom I've known since they were babies and are now in their 20s, my nine-year-old daughter, and my stepchildren's two younger sisters from their mother's second marriage. The two younger sisters would visit their siblings (my SD and SS) at our house sometimes. We got to know them really well. At one point, because their parents were in a very nasty drawn out divorce, we offered to have them live with us. Sounds crazy, maybe, but our offer was actually considered, but in the end not accepted.
We have a family picture of all five kids together and my daughter counts that she has one older brother and three older sisters. I don't think kids can ever have too many adults who care about them. I hope this mother can let go of some of her anger and see that. Keep offering.