tl; dr: You should tell the travel agency the full name. It's probably okay if you do not, but why would you be happy with that?
The names of airline tickets are often a bit strange written (z LAST NAME, FIRST NAMETITLEand, realistically, chances of you encountering problems if you ignore or abbreviate the middle name are very close to zero.
But other than that, you'll gain absolutely nothing (except, I think, bother typing your fiancee's middle name – which is far less than you've already asked this question) and without knowing all the details of the planned trip We can not be completely sure that you are not flying with an operator or a target that has the correct (complete) name does Make a difference. Tell the travel agent the full name and you do not have to worry.