Pacific, I think you are in a bit of a hurry, with both the skin issue and his weight. 4 weeks is bugger all. 
Don't expect to see changes overnight, concentrate on just getting condition right, and the weight will follow. Personally I wouldn't feed Burns (there's nothing wrong with it) but a lot of dogs stay lean on it. Some greys (we have one ATM) are hat-racks by inclination so you need to work extra hard on the feeding, and it may well be that he will always be lean. As long as he is healthy and fit, and your vet is happy, that is what matters.
Skin - are you grooming him every day? The act of grooming is very helpful in removing scurf and getting rid of dead hair and encourages circulation. I would personally also add a spoonful of olive or sunflower oil to the feed. With time and good feeding this will also come right.
One answer that may help both is to do supplementary feeds of satin balls. Yes, they really are called that! There are many recipes kicking round on the internet, but the basic idea is that you make little balls or patties which are nutrient rich and feed them as a supplement. They help with both weight gain and skin condition.
You may also find it helpful to break his feeding down to three or four smaller meals each day and in this way, you can get a higher total of food in than if you are just feeding say two big meals. Also don't underestimate the value of high value treats for both improving and working on recall, and giving him extra calories.
Raw is great, but at this stage, would you be introducing too many changes too quickly?