Every rational relation is computed by an automaton with output and extended transitions in which every accepting run is in ε-free normal form: if the input string is nonempty, each transition reads exactly one letter, and if the input is empty, the run has exactly one transition (Lemma B.2.4 of Transducers, first sentence). After arranging that no state is both initial and final and that every transition reads at most one letter, the usual elimination replaces, for every letter $a$ and states $p, q$, the runs from $p$ to $q$ reading $a$ and any number of empty-input transitions by one transition labelled with the regular language of their outputs; a fresh initial and a fresh final state with one transition between them, labelled with the outputs on the empty input, take care of the empty string.