Troffinch is an unevolved pixpet, whereas the pixpet you are fighting is evolved. Evolving your pixpet gives her access to better scrolls that do more damage and can reduce accuracy. The attacks you're using are beginner ones, where as for that quest you definitely want heavy hitting and status moves. Evolving also gives your pixpet better base stats, which means it will get stronger, more defense, a larger health pool, and more stamina.

There is a bird on Unity Bridge that sells better scrolls for bird-types also!

Good luck