Hello and welcome to AADF forum. Yes, we have had several threads about coprophagia - it is clearly a common problem. I was just looking at the Burns website earlier today and in particular the Burns Health Guide
. You seem to be associating the coprophagia with diet so have a look at Page 21 and see if there is anything useful in the article.
I honestly don't know whether it is diet related or not but I have formed the impression that it is an unfathomable habit that the dog will persist with whatever it is fed. I reckon the owner just has to deal with it in a practical way - clearing poo straight away and keeping the dog on a lead (or muzzle) when out and about.
Regarding the food and protein content, you don't say why you think that Barking Heads is not suiting him so maybe you could let us know? There could be something in this that would help you to eliminate some other products from the list of possibles.
There is a trend now for higher protein foods, particularly raw meat. Dog owners have to decide if that is the way to go for them. I am just a dog owner of many years standing but I feel that the average domestic dog may not actually require this kind of diet and that a suitable protein level should be related to the dog's activities of daily living. However, I realise that some people don't agree with this.
It seems that you want to remain grain free because of the seasonal allergy and of the foods that you describe, the only one I have personal experience of is Wainwright's. I gave it to two of mine fairly recently and I think it is a suitable product for the average domestic pet. My two had perfect poos on it and there is the added benefit of being able to ring the changes on meat source. The protein is moderate at 27% and the carbohydrate source is mainly sweet potato which is said to be quite good. One or two people on my own forum have put their problematic dogs on Wainwright's grain free with good results. The other grain free product that I used for quite a long time was Fish4Dogs. All mine were fine on it and I cannot remember why I took them off it - maybe it was just for a change.
Edit: While out on a dog walk just now I was thinking about my friends who have owned and bred Labradors for many years. They have five between them, two being of similar age to yours. For some time they have all been on white label duck and potato with a protein of 26% and fat 12%. They are typical, laid back Labs and don't get (or require) masses of exercise so these levels are OK for them - they are all doing well and look good. However, one of them is a persistent poo eater and has been since she was a tiny tot - wouldn't matter what they were fed, if the poo is there, she'll eat it.