I think you're reading a bit too into this, when I first read this, I hadn't played through the scene yet and it got me really worked up. For one, Cooper only asked Cyrus to turn around because he felt uncomfortable changing in front of someone he didn't know (given kind of weird that you don't mind being naked in front of someone you don't know but honestly, people be weird like that, as in they are comfortable with their own body enough to not care). For two, from the transition we can extrapolate that a conversation was had about it, we just got to see the tail end of it, so Copper had plenty of opportunities to flat out say no to Cyrus's assistance. For three, his 'reluctance' is more like a kid begrudgingly doing the homework that isn't graded, he doesn't see the point to a bath because it doesn't bother him.