Hit em with a frenzy spell within a town and let the guards do the dirty work for ya

You could also make them a Steward of a different house, one you don't typically occupy. (I usually keep the other two Hearthfire Houses at Small House layout for this purpose)

Mace of Molag bal..... oh wait... that's for Serana.

This ^

But does the OP want to get rid of her permanently? If yes, Boethiah or Mephala are the answers.

Pretty sure I shipped three of my first characters annoying housecarls off to the Blades, Lydia always went as a steward of a Hearthfire house and he married Iona. My original character has no housecarls cluttering up his vanilla homes and making annoying comments.

The second character who used three for Hearthfire stewards, I think still has two, one of which she will probably eventually marry. I suspect the second character still has Iona lurking around Honeyside. Getting rid of annoying housecarl's is not enough incentive for that character to go do the main quest. And Iona is in no danger of dying. A less tolerant character might have sacrificed her on some alter.

I took Lydia with me as a follower thru Labyrithia on a quest, and she didn't make it through the penultimate fight. I certainly didn't mean to kill her but she could have picked up a bit of an AOE spell while kneeling. I can appreciate the OP wanting to get her out of Breezehome -it is creepy having her in ones bedroom and that house is small.