To give you perspective from the 'other' side, I can say that my DS could have been your DSS. I have spent all his life pestering him about showering, brushing his teeth and keeping his room clean. He is a wonderful boy, intelligent, kind, extremely mature and independent, always praised by his teachers, but when it comes to hygiene....well, somehow, that doesn't seem to come so easily.
When he was young, I used to buy him his clothes without him with me and he wore whatever I bought him. Then I started to notice that he wouldn't wear a number of items and he admitted that some things he didn't like, so I told him that he would need to come with me. I might as well have told him that he couldn't watch TV any longer! He absolutely hates going shopping. Last year, after I pestered him over and over, I said that he had no choice but to go. That left him in tears telling me that I was ruining his day! The prospect of spending my limited disposable income on clothes that he didn't want wasn't a pleasant experience.
He is now 13 and I think just about turning around. I still have to remind him to shower and brush his teeth, to go and get changed when he puts dirty clothes but at least I'm not getting a grumpy response back any longer.
As for the 'he doesn't it when he is with us', well so did DS, but that's because he felt much more intimidated with his father and SM than at home.
The reality is that if he is coming to the age of 10 and he does care to be clean and properly clothes, he should be starting to do these things of his own accord without being reminded, so maybe things with his mum as not as they seem and maybe you shouldn't be jumping assuming that it is all the fault of the ex.