Interesting question.
It's all in the spelling and how many "e"s are in the word. My dictionary defines fiance as "a man engaged to be married", and fiancee as "a woman engaged to be married".
Our fraternity is already gender based in that we don't initiate women. And as Grits and Ottor pointed out, the Code is also gender based in allowing only female non-members to wear the badge (mothers, not fathers; sisters, not brothers). Therefore, limiting the wearing of our badge to female fiancees (and not male fiances) would be consistent with, and no more discriminatory than, the policies we've had all along.
The reasons should be obvious. A man wearing our badge is assumed to be an initiated member while a woman is not.
Furthermore, how would it look if the significant other of a gay brother was already a member of another fraternity? Would he wear his Lambda Chi badge a little lower than, say, his TKE badge? Oh, the horrors!
So in my opinion the answer is no.
As for legalities, I've never heard of any lawsuits brought by fathers or biological brothers just because they weren't allowed to wear our badge while their wives or sisters were.
Notice that all these statements refer to wearing the badge as opposed to simply having one. Many badges are in the possession of male non-members (such as sons and grandsons of deceased Lambda Chis, antiques dealers, and collectors), although they don't wear them.
As Ottor pointed out, there's no "Badge Enforcement Division", so no one is going to break down your door for giving a badge to a loved one as a token of affection. In my opinion it should be acceptable for a father, brother, or fiance to be presented with a badge, but improper for any of them to actually wear it. They could, however, wear a friendship pin, which would be a good alternative when attending events.
Does this make any sense?
ZAX,
Jono