I'm currently implementing my own trie (for learning) for my own autocomplete module ... and I don't see how a trie (prefix tree) can solve the issues you just wrote.
When you traverse the prefix tree and you are blocked: meaning, the prefix does not match an entry in the structure, you perform edit operations: insert, delete, transpose, and substitute. If one of those garners a match, you keep going down the tree until you reach your edit distance.