Disclaimer: I am not sure if this is the case for your poor Sidney, nor do I know if kidney stones feel the same for dogs as they do humans. What I can tell you is that, when I have had kidney stones, in addition to the pain from the stones, I always felt like I needed to urgently urinate. All the time. I would literally walk out of the bathroom feeling like my teeth were floating, even though I just went. This was particularly true when I would change positions from sitting to standing, from standing to laying down, etc. I wonder if this is the case for Sidney too. Perhaps his body is telling him it is urgent and that is why he isn't asking to go out. Would it be worth trying a belly band for a while? I hope you can find a solution that works for all of you!