Sadly there aren't many around any more. I tend to GJW Titmuss own brand personally. I know that link says not to go too low on protein. The key here is animal protein. An animal that is omnivorous maintains a good weight on diet like rabbit muesli when in older age. I've got some rather chunky male rats even on a plain diet like this!
Research into specific hamster issues is thin on the ground at best and much of what the vets know is taken from other species plus what works. See how you go with the diet tweak and go from there
I'm not certain, from testing urine at my previous job at the vets, that you'd get enough for a proper urine test in any case but it might be enough to rule in or out protein in the urine at least. You can buy test strips at the chemist I believe or online for example (you'd be looking at the multistix)
https://www.viovet.co.uk/Bayer_Urine...ck_Tests/c789/
You can get your hamster to pee in a clean plastic box and suck up the urine in a syringe. This way you can pop a drop the pads of the multistix you need. But yes, I reckon a specific gravity test for a robo would be asking a bit much
Process of elimination. If changing the diet helps it might still not be kidneys but it would definitely short list it as highly probable