One possibility for Fanny's queer hairstyle could be that she has it cut in a short layered crop - a style called "A la Titus" or "A la Brutus".

The style originated in Paris during the Reign of Terror (c.1793-4) and echoed the cropped hair of women going to the guillotine. Known as "A la Victime", this style was cut very short indeed at the nape of the neck, and was sometimes worn with a thin red ribbon around the throat. (Lovely, eh?)

Slightly softer versions appear in illustrations in French fashion magazines in the late 1790s - see the link at the foot of this post.

The women's style "A la Titus" or "A la Brutus" was not dissimilar to fashionable Regency men's hairstyles, which were modelled on images from classical Greece and Rome.

Some women continued to wear their hair cut short like this through most of the Regency period. One example is Lady Caroline Lamb, nee Ponsonby, who became passionately besotted by Lord Byron in 1812 and famously described him as "mad, bad and dangerous to know".

Here she is dressed as a pageboy.

And here's another image, c.1820.

This hairstyle suits her very well IMO, but on less delicate features it would not be at all flattering. Perhaps this is why William thought the ladies at Gibraltar were mad.

Sources include:
"Passion & Principle - the Lives & Loves of Regency Women", Jane Aiken Hodge, 1996.
"Lord Byron", Peter Brent, 1974.
"The Gallery of Fashion", Aileen Ribeiro, 2000.
Cathy Decker's "Regency Fashion" website.