I know I will be shot down in flames for saying this but it's Friday and I've already had a glass of wine so here goes...
My sons name is a nickname and I went through the same quandry. I don't like the formal version of his name, nor do I believe he would ever want to use it, so we have stuck with his name (the nn) on his birth certificate too. When I was fretting about this two things were pointed out to me which I tend to agree with...
In 20 years our children will be part of a generation where nicknames without formal names will be part of the norm. It will be a generation of Charlies, Millies, Alfies and Evies and what seems awkward or cutesy to us now just won't be by the time they are adults. Whilst I agree that having the option of both might be more flexible I don't believe our children will see it as necessary, particularly if you don't like the longform name in question.
Secondly, and this is where I know I'll be in the minority, I don't believe that the lack of a formal name will in any way hold your child back for the future, or hinder them from becoming say a Judge, Politician or CEO. Willie Walsh, Tony Blair, Archie Norman, Nick Clegg have all done very nicely. Yes, I'm sure they may have called themselves Anthony, William and Archibald on their CVs but the fact remains that it's their nicknames which they use, and we're quite comfortable with it. Whilst we may from time to time question their ability to do their job it's unlikely to be because they called themselves Willie not William. I would imagine the name Cherie may have got pretty short shrift on the name boards but Cherie Booth/Blair clearly doesn't let that bother her as she chooses to use it professionally over Theresa (which is her formal name).
Lexi, to me, stands up as a name in it's own right. I see it as Jake is to Jacob. A name that derived from another but which is perfectly acceptable independent of that. If you love it go for it. I'll look out for a Lexi ruling the world in years to come and hope she's yours!