Update. It is undecidable. Here is the proof.
If $f,g\colon A^*\to \{a,b\}^*$ are two morphisms, then one can construct a rational Z-series over A whose support is the complement of the equalizer of f,g. This is how Post correspondence is reduced to universality of Z-series and is based on a faithful 2x2 rep of the free monoid over N. See the proof of Thm 27 of http://www.infres.enst.fr/~jsaka/ENSG/MPRI/Files/References/JS-HWA.pdf
So it suffices to prove it is undecidable whether the equalizer of two free monoid morphisms is rational.
This is shown undecidable in Thm 5.2 szeged.hu/actacybernetica/prevvols/4_1/4_1_127_139.pdf Thm 5.2. It is also shown undec for context-free.