I believe your specific question was that you wanted information on what could be done...
The best answer to that question is to find out what is causing the problem. You need to continue looking for the reason behind these symptoms, then by treating the underlying problem, hopefully the symptoms should resolve.
The enlarged heart finding on the xrays may or may not be significant. The way to know for sure? You should take him to a cardiologist who can do an echocardiogram and see if there is a decreased function to the heart, or problems with the flow of blood through the heart. Right sided heart failure will cause the liver to enlarge. Who knows, maybe his heart is making him weak and causing his walking troubles and panting, andcausing the enlarged liver. Since the liver enzymes are normal, that would indicate the liver is functioning normally. If this were my dog the echocardiogram would be the next step.
You could also look into whether or not the neurologist can do an MRI of the brain looking for problems there. Unfortunately, I don't think you have gotten to the root of your dog's problem, and more testing will be needed. Sometimes problems aren't identified on the tests we do routinely. The only thing you can do, if it is in your budget, is get further tests done. Ask your vet and also the neurologist what they feel the next step is. I'm more inclined to check the heart. However, if the liver tests you are supposed to be getting are abnormal, then a liver biopsy might be next!
Please reply with any further information or questions, I'd be happy to keep trying to help.