Likely apocryphal. It isn't in the massive official "Despatches of Field Marshal the Duke of Wellington" and the exaggerated, humorous style is not characteristic.

Thanks, I read through a few pages and found it a more interesting read than the original link.

E.g. here he's complaining to the undersecretary of state:

https://archive.org/details/dispatchesoffiel10welluoft/page/...

Here (and a few lines on the page before) is a long letter with his advice on how to reconstitute the (allied) government of Spain:

https://archive.org/details/dispatchesoffiel10welluoft/page/...

Also, the letter is allegedly (1) dated 1812 and (2) signed "Wellington". In 1812 he was still plain old Arthur Wellesley; he wasn't duked[1] until 1814.

[1] I am sure this is not actually the right term. I do not care.