I would guess it is a highly individual thing - the Duchess of Devonshire, for example, called her husband 'Canis.' I can't recall what nickname she had.
As for Darcy, he doesn't strike me as the 'nickname-y' type either, but then, Fitzwilliam is a mouthful.
I could see Elizabeth taking her cue for what to call him from the rest of his family, but since there are no clear rules, your guess is as good as many and I thought that it worked for Elizabeth to call him 'Darcy' in your story.
As for Darcy, he doesn't strike me as the 'nickname-y' type either, but then, Fitzwilliam is a mouthful.
I could see Elizabeth taking her cue for what to call him from the rest of his family, but since there are no clear rules, your guess is as good as many and I thought that it worked for Elizabeth to call him 'Darcy' in your story.