score:2
tl;dr: You should tell the travel agent the full name. It's probably fine if you don't, but why settle for probably?
Names on airline tickets are often written in somewhat odd ways (eg LASTNAME, FIRSTNAMETITLE) and realistically the chances that you'll face any problems if you ignore or abbreviate the middle name are low.
But having said that: you gain absolutely nothing by doing so (except, I suppose, the effort of typing out your fiancee's middle name - which is far less effort than you've already spent submitting this question), and without knowing all the details of the trip you're planning we can't be totally sure that you're not flying with an operator or to a destination where having the correct (full) name does make a difference. Tell the travel agent the full name and you won't have to worry about that.