So this is kind of JAFF related because in P&P no one calls Darcy anything except Mr. Darcy. But as his first name is Fitzwilliam, a lot of people imagine Elizabeth calling him something shortened, like Will.

Here's my issues. Firstly, people are constantly saying Mister Bennet and Missus Bingley. They are used to saying long names. So they may not feel the impulse as much as we do to shorten. Most nicknames are used to distinguish people from one another (Like Fanny since her mom is Frances). One of the only people in Austen who shortens his kid's names is Mr. Price in Mansfield Park, who calls Susan and Fanny "Sue and Fan", but he is vulgar.

Secondly, Darcy doesn't stop being proud of his family/heritage/name, he just gets it under better regulation. He is proud of his family heritage! His first name is the name he shares with an Earl! That's why it's there! I really doubt he wants people to shorten that part of his name. He would probably find it offensive.

If anything, I can see Elizabeth calling him "Darcy" but not a diminutive of Fitzwilliam.

Last point: Elizabeth is referred to as both Eliza and Lizzy, but the narrator, who is mostly from her perspective, always uses Elizabeth. Which has always led me to believe she prefers her full name. Fitzwilliam and Elizabeth are basically the same length. They can handle it.