how long is he doing this?What does he eat?
does he have a lot of mucus that causes post nasal drip?
Bulldogs and other Brachycephalic (short faced) breeds often suffer from acid relux. Many times they have post nasal drip complicated by a short trachea or palate. There is a condition called hypomotility where the food is not moving quickly enough through the digestive tract. This is more common at night when the body slows down.
If he only has this problem at night give him a half a pepsid tablet ten minutes before he eats his nightly meal. Do not let him fill up on liquid before, during or after he is eating. Let him only have an ounce every 10 minutes until an hour after his food. Walk him before he eats so he is not subjected to any exercise afterwards. You can increse the pepsid to a full tablet if half does not completely resolve the problem.There is also a vet scripted drug calle sucalfrate that he can try which is like canine prilosec. He is on an excellent diet. Both Eukanuba and Royal canin should provide excellent nutrition and are low allergy as well.
Please let me know how he is doing on the pepsid. If you have any more questions, I wil lbe happy to discuss this further and help in any way possible.