Every function computed by a two-way transducer is regular, i.e. decomposes into rational functions, map reverse and map duplicate (Theorem C.2.9 of Transducers, the hard implication). The run of the transducer is described by its snake graph, computed by a rational function, and the output of a snake graph of width at most $k$ is a regular function of its representation (the snake lemma, Lemma C.2.12), by induction on the width: a snake of width $k$ is cut, at the record-breaking columns, into looping and progressing parts of smaller width, whose outputs are computed by the induction hypothesis on factors cut out by rational functions and glued back by the closure properties of Lemma C.2.10. A run that halts visits every column at most $|Q|$ times, so the transducer's function is its own width-$|Q|$ snake output.