It's not personal. The boy would hate you whoever you were. Don't cook or launder for him unless it's absolutely no inconvenience. His dad can do it! Or you can show the boy how to care for himself.
If you imagine the boy to be someone beyond your responsibility, especially in terms of developing his manners or deciding what he must do next, then you will be free to do as you please. Go out, take a bath, view the city parks/art gallery/book shops or have friends round. Spend money on yourself if you've got it.
Can you open your mind to forgive the sullen teen for being confused and nasty, and plough on as if it's water off a ducks back? He won't be a teen forever. He wants to be with his dad and the situation isnt ideal. Who knows, your relationship with big lad might warm up. It takes two to tango and you can't force it, but you can be there in the wings if he wants to try to connect
Him accepting you warmly might be beyond him at the moment, as if it's a rejection of his mum (maybe) or a jealousy of Dad's attention, or because he's a selfish teen who's self centredness means he would treat anyone he wants to this way. One day he might be able to see how it is for his step mum and more likely for his Dad.
All you can do is your best. Smile even when your heart is breaking, and do your own thing. It will become the new norm.
You might find they want you to be involved with them a little more if you start looking like there's no power struggle of passive tormentor and injured adult.
Fake it to make it. And try to be kind to yourself. It will brighten your mood if you compensate for the boy's passive aggression by treating yourself nicely, the way we all deserve to be treated.
He cares less about the situation that is intolerable to you and is muddling through without as much angst as you, even if he does mouth off to others. His dad probably is at a loss about knowing how to help, otherwise he would have put his foot down tot he rudeness earlier.
Start a mission to try to say something nice and brief to the lad each day, with a smile, and then get on with your own thing. You can pave the way for when he matures and he might soften.