According to the obituary of d'Éon published in 'The Gentleman's Magazine' after their* death, in May 1810, they were 'attended by Pere Elisee' in their last illness. 'Pere' or Father Elisee would have been a French Catholic priest, of which there were many in London at this time, having fled Revolutionary France in the 1790s. The presence of a priest strongly suggests that the Last Rites were administered, although I can't find a source which confirms this. Although d'Éon was buried in St Pancras, London (a Protestant Anglican church), it was a popular burying ground for French Catholic émigrés, and undergoing Extreme Unction at the hands of a priest would not have precluded d'Éon from being buried here. (See here for the full obituary: https://tinyurl.com/yblvlufe)

*I'm steering clear of his/her gendered pronouns, as d'Éon's own gender identity was not clear in their lifetime.

EDITED TO ADD: In the Catholic Church, the Last Rites and a funeral are not mutually exclusive... I assume you were asking whether Catholic Rites (plus funeral) OR a Protestant funeral were administered?