What's a good word to describe someone who is prone to sudden changes of mood?

Perhaps mercurial. My dictionary (Chambers 13 edn) includes

active, sprightly, often changing

in its definition of the word.

In former times a woman, such as in the example sentence, might be described as flighty, which the same dictionary defines as

swift (Shakesp), fanciful, changeable, giddy-minded, irresponsible, flirtatious

I suggest volatile

likely to change suddenly and unexpectedly, especially by getting worse

or

capricious

showing sudden changes in attitude or behaviour changing suddenly and quickly

or

erratic

moving or behaving in a way that is not regular, certain, or expected

She was a volatile / capricious / erratic woman, a woman who was prone to quick outbursts and fiery changes of mood.

moody adjective
2 : subject to moods : TEMPERAMENTAL
Merriam-Webster

You could call such a person tempestuous, if by 'fancy' you mean 'literary':

characterized by strong and turbulent or conflicting emotion (Oxford Languages)

1] of, involving, or like a tempest 2] violent; turbulent (Collins)

Another good word is temperamental:

marked by excessive sensitivity and impulsive mood changes...unpredictable in behavior or performance (Merriam-Webster)

From those definitions, turbulent also works. So does fiery, which you used, and hot-headed, although the latter isn't quite so elevated.