Complete resolution rule for 1-in-k SAT
You're treating resolution as if it were a purely syntactic rule.  It works that way with traditional CNF clauses because that corresponds with the underlying rule of inference.  But a CNF clause with the added restriction of only one literal allowed to be true no longer corresponds to what the resolution rule can be validly applied to.

The Boolean expression $(a \lor b) \land (\lnot{a} \lor \lnot{b}) \land (\lnot{a} \lor b)$ is unsatisfiable as a 1-in-k-SAT formula but naively applying the resolution rule produces $a = false, b = true$ as a (wrong) solution.