Deathless
Let’s just assume Thrain spent a good decade or three more on his wanderings before being killed by Sauron.
On one of her adventures Belladonna Baggins has a torrid affair with a batty old dwarf who claims he is the rightful king of some long-lost mountain. It was jolly good fun, but she has a smial waiting for her back home with her husband inside. Bungo, bless his heart, takes the news of her pregnancy by another man, err- dwarf, in good stride.
And when their son is born, well, he has Bungo’s chin, doesn’t he? The neighbourhood certainly doesn't need to know that her Bilbo isn't a Baggins, let alone an actual Hobbit. And his feet are furry (just slightly so) and his ears are pointed (just a bit rounder than normal) and he eats like any Hobbit would.
It's just that sometimes, when he corrects her on her history or spends hours weaving the tale of some mountain kingdom and manages the other fauntlings like he is some sort of princeling, that Belladonna thinks that the only odd thing about her son is how he seems old beyond his years.