It does sound like a behaviour issue rather than food. I would be inclined to use a relatively low protein food for such an active dog, stops them from being OTT. It is really a good idea to get the best quality food you can afford, especially if there is a sensitive tummy. I swear by James Wellbeloved, it is hypoallergenic so really good for any food intolerances. It is on the expensive side though.
Does he do this at specific times? Or just when no one actually around. The key i think would be to stick him outside when you are working around the house etc. A brilliant tool is a crate, dogs never want to soil their bed, so when he is mooching around and lazing, if he is in a cage he wont shit in there. It is a brilliant training device and place of retreat for a dog.
ARe there any stress issues that have arisen? Does he do this if you have been out, separation anxiety?
I'll let you into a secret, i'll get flamed, but it worked for me, my wee terrier (and my rottie before him) i rubbed his nose in it. Im not recommending you do this though as if there is an anxiety issue you could make it worse.
Again, with the crate, if he messes up (literally) straight to the crate, ignored.