It sounds as if you are having real problems with your Labrador. Finding the trigger factor for allergic reaction is never very easy and it may or may not have something to do with food. At least your vet is onto it and you are having tests done.
In answer to your question about which dog food is best, I can only give general advice because some of the best rated dog foods on here i.e. those that score between 4 and 5 stars, may not actually be suitable for your particular dog. Some of these (particularly kibble) have an awful lot of ingredients in them and it makes it difficult to identify which might be causing the problem.
As Tinyplanets has said, you could use the filters on the Dog Food Directory of this site to narrow down the choice but here are a few pointers which might be helpful:
* The commonest allergens are said to be chicken (and poultry products) and grains so it might be best to look for food that does not contain these.
* Consider giving your dog a wet diet. They tend to have fewer ingredients so less to cause problems. Also it eliminates the risk of storage mites that can occur in dry food - dogs can be allergic to these and to their excreta.
* Choose one protein source and stick to it. Introduce other types gradually so that you can monitor the response.
* Check out the review of Chappie on the Dog Food Directory of this website and you will see that it is not highly regarded. Probably one to avoid for your Labrador
* Excessive thirst when using dry food - as Tinyplanets has said, this is normal in kibble fed dogs. Kibble can be soaked for a while before being given and this should reduce the need for the dog to drink so much after Soak some kibble in a clear glass of water and you will see how it expands. It then becomes obvious why the dog is so thirsty.
Although you have discounted a raw diet, you don't say why. I think that it may be worth thinking about because it is plain and simple - just meat, offal, bone and vegetables. However, some vets don't approve of it so if yours is one of these you may be put off. There are some great complete raw meals on the market now - again, check out the directory but a few are: Natural Instinct, Nutriment, Natures Menu and Honey's. All of them have customer service staff that will be able to help you find something suitable. I was recently told that Honey's will do individual recipes on request.
Please let us know how you get on with your Labrador and if you need any help with the Dog Food Directory filters/settings, please ask.