Like you say, he's probably behaving better for you guys because you have boundaries. It's the same with my DSD6. She's an angel when with us but apparently can be a nightmare at her mum's. We have strict routines and boundaries whereas things are more relaxed at her mum's.
I don't really think you need to do to much as it sounds like you're already doing a good job. Its the mum who needs to tighten her boundaries.
I would sit down with DSS and discuss some rules. Let him come up with some so he feels involved and like your not just telling him what to do. With my DSD we asked her "now that you're 5 what chores do you think you'd be able to do?". My DSD had a reward chart when she was younger and now has a GoHenry card. She gets her pocket money on a Friday if she's done all her chores that week and she can make extra money by doing extra chores. Her daily tasks are making her bed, opening her curtains, getting herself ready for school, putting her washing in the basket and taking everyone's plates though to the kitchen after dinner. I think when you give them a bit of responsibility they behave more responsibly.
Before he begins going back to his mum's, I'd suggest sitting down together with mum if you have that sort of relationship and discussing the routines and boundaries at your house so you're all on the same page. So "DSS has been getting up for school on time, he goes to bed without arguing, he tidies up his toys etc.
I get his mum is at the end of her tether but honestly, I don't think sending him away is the answer. The issue is clearly with her parenting rather than the child. He's eventually going to have to return to her and unless she makes some changes his behaviour will remain the same.