...to make himself available to corroborate Darcy's story if Lizzy requested it. Fitzwilliam even considered going out to search for her.

I do not think Darcy told Fitzwilliam of the proposal, but he easily could have said that Lizzy was vastly mistaken in Darcy's treatment of Wickham, that Darcy had given Fitzwilliam to Lizzy as a reference who could vouch for Darcy's honorable behavior, and that Fitzwilliam should feel free to answer any question Lizzy might put to him on the matter including those of his sister.