Judging from a few other posts on Mi Yodeya I am left with the impression we do not say anything for a deceased pet. But asking just to be sure.
According to this site, the answer is superficially yes, but according to one interpretation, no, as a pet is not considered a monetary loss.
This answer seems to point to its being an OK construction for all sorts of sad moments.
And a comment on the Chabad website reads that the proper thing to say is "May the Omnipresent One make full your loss (HaMakom Yemalei Chesroncha)" -- see Tractate Berachos 16b".
According to SA (OC 222:2) Baruch Dayan HaEmes is said on any bad news.