Usually good quality wet foods have a higher fat level which should help your dog to gain weight. However, they tend to have a correspondingly higher protein too and you have indicated that your dog does not do well on high protein.
The product you have chosen, Barking Heads Bad Hair Day has a moderate protein of 25% and higher fat, 18.5% so in theory should be a good choice for your dog. The review of Barking Heads Bad Hair Day is here.
The brown rice and oats will add carbs and fibre which should help with intestinal health. It would probably be better if you gave nothing but the Barking Heads so that you can evaluate it properly. He should not be hyperactive on this because the protein is only 25%.
I don't see probiotics in the formula so if you want to supplement, have a look for something or maybe ask your vet.
As his poo is now firming up I would advise you to stick with this product. Don't forget that you can soften the kibble with warm water to make it easier to digest. Also the dog will be less thirsty. Weigh the food and don't be in a rush to get the weight back on. It's important that you don't overfeed. For now, concentrate on stabilising your dog rather than gaining weight. Once the poo is nice and firm you can very slowly increase the amount if you feel it necessary.
Going forward, Wainwright's grain free might be suitable, particularly as there is a flavour range. However, I really think that for now the best thing is to stick with the product you have chosen as it appears to be helping.
Your little dog has had a tough time and I hope that he is soon better. Please would you let us know how he gets on?