Hi I'm Rosie one of the vets and I'd like to try and help you. I'm sorry your dog is in so much pain - it is awful to have to see them like this.
To be honest, what you are describing is concerning to me. For most straightforward pulled muscles tramadol and rimadyl should have helped - for most sprains for back pain rimadyl only is usually enough for patients to be significantly more comfortable. I do agree with your vet that back pain is the most likely diagnosis, but I am not entirely convinced that it is a simple pulled muscle.
It could be that he has disc disease (I'm not sure of his breed but some breeds like Dachshunds
are more prone to this, but i have seen it in a wide variety of breeds). This is where the disc (almost like the cushion between vertebrae) sticks out of where it should be and into the spinal cord. Dogs can do very well with this - sometimes they need surgery and sometimes strict cage rest will do. He could have a fracture somewhere (unlikely without trauma, but possible). He could have in infection in his bone
(which can be extremely painful). I also have to mention that some bone tumours are incredibly painful - I don't want you to panic, but you need to know this is a possibility.
It could be that your vet has misdiagnosed completely and he has abdominal pain, but I would expect him to be vomiting with this, and it wouldn't make his back legs uncoordinated.
At the end of the day, if I saw him at my clinic and he wasn't responding to medication then I would want to investigate and see why not. The list of possibilities is quite long, as you have seen. I would ask your vet if she do further investigations. The first one I would want to do would be an x-ray of his spine and the area of ribs that are hurting (they can have referred pain whereby a spinal problem causes pain in the ribs, so I wouldn't entirely discount the spine). If that was normal then he would probably need more detailed imaging of his spine - either a CT scan, or MRI (both which usually needs referral), or a myelogram. This is where dye is injected into the fluid around his spinal cord, and you (or your vet) can use it to see what his spinal cord itself is like.
If the vet can't, or won't, do further tests then it might be time to seek a second opinion from another vet. Like you, I am worried that something more is going on here and you really need to find out what it is before he can be treated properly.
I hope I've helped - if you have any questions from this then just hit 'reply' and I will get them and get back to you. Otherwise, if you have found this useful, please leave a positive rating so I may be compensated for my time. Thank you, ***** ***** luck, Rosie.