NNs are not a must, and I leave it completely up to the child. If you truly loathe
Alex, then I would avoid it because
Alexander might grow up wanting to be called
Alex. However, if you just prefer
Alexander, calling him
Alex is not a must and you can just make it clear to family his name is not
Alex.
My cousin
Anthony has never been called anything but that. I also met a few kids when I worked at a camp that never went by anything but their full name. Among them were
William,
Daniel,
Christopher, and
Zachary.
NNs are not a must and I don't understand why people avoid a wonderful full name so their child's name is not shortened when its up to them to tell people its just
Alexander (in your case). I love lots of names where I don't like the nns, but that doesn't stop me from using them, unless I truly hate it (for instant, I dislike
Will, but not enough not to use
William, but I loathe
Lilly and therefore would probably never use
Lillian despite it being a name DH and I agree on)