I would call this character building, both in practice and to the editor. The narrator is in Bilbo's mind (to know he is bewildered) and **Bilbo* considers himself "bewuthered", so this is a word he learned, perhaps a colloquial expression. I would assume, since "bewildered" indicates confusion, that "bewuthered" indicates something else, like frustration or irritation.
I would coin words sparingly, but in particular I would use coined words to highlight something about local culture, in this case, Hobbit culture. For myself they would refer to invented social roles, invented social customs, invented holidays or events or milestones in a life, like formally becoming an adult, or candidate for marriage.
They emphasize to the reader that these characters do have their own upbringing, thought patterns, and are not the neighbors next door. This is part of the fun of the fantasy, these hints of a whole other society and customs. But it can be overdone, I think it should be limited to usage where the meaning is easy to guess by context, or the meaning is not critical to the story. Both are true for "bewuthered", we don't know what it means precisely but we can guess the general tone of the meaning, and the exact meaning is not critical. In other circumstances, the meaning can be more precisely conveyed:
"Wait, aren't you betroken?"
"No, I lack two months."
"Seems like longer, I guess."
"To me too, believe me, and if I was, he'd get an earful from me."
So you get the idea "betroken" is something like tenure in this context, it provides some kind of immunity by seniority.
I think this usage by Tolkien is a valid candidate too, it sounds like some of our own couplets commonly used in thinking, "sick and tired", "said and done", "through and through", etc. But it rhymes a bit, like "tip to tail".