mountaineer female/woman

[Tdol]

I've also found this one: "She was a typical mountaineeress, tall, angular and sallow, but there was gleam of intelligence in her face, quite unlike the lacklustreness of the usual woman of the mountains." Link

That's from the Athens Reporter back in 1894. Google Ngrams couldn't find mountaineeress in its huge database. I'd drop that word. There are plenty of instances of mountain woman, and there have been for a long time.