Password derived hash to encrypt known plaintext as password check
To be entirely honest the Idea is pretty intresting but wouldn't help a lot. AES is, mainly because it's widely used, a pretty cheap thing to do, so there's basically just one step more after hashing, so it wouldnt make a lot of a difference in that part. but one thing that I think may be intresting would be (but for a different reason):

Not encrypting a static String, but rather use the hash to create or encrypt a random Private Key generated while the user is signing up, while keeping the Pubkey readable and offer it to the user.

If you now use that specific key to sign something (similar to a canary report) and the user can check that it was signed with his key, then he can be relatively sure that for example no password reset happened because one cant reset the password without resetting the keys.