Not on the old Fandom platform. However, this should be possible on UCP. Go to the Special:Version page on your wiki, and if the version for MediaWiki is listed as 1.33.3, the wiki uses UCP, and you should be able to move a category.
With a dev script, yes. But you’d need an account for that. And if it’s a big category, preferably also a bot account.
What do you think?