Is it possible to tell Google's Webmaster Tools that our site hosted on
www. has moved to a sub folder, sub domain or a new domain with sub folder?
The tell Google you've moved feature is designed for old domain to new domain and does not support sub domains, sub folders or new domains with sub folders, it is strictly from domain to domain.
This question has been asked various times on many forums across the internet, Google does not treat a domain to sub domain or folder a site move.
You should proceed with a standard 301 which is more than enough to tell Google that your content has permanently moved.