You might try a hepa filter in the room they sleep in overnight.
If they are doing a lot of coughing in the morning only I would be concerned about possible airway issues due to allergies from where they are confined overnight.
If this coughing and choking is really new and they are coughing some during the whole day too, just more in the morning, they might have picked up kennel cough or the new canine influenza.
www.marvistavet.com/html/body_kennel_cough.html
www.marvistavet.com/html/body_canine_influenza.html
Because this breed is known to have serious airway issues and a problem with cooling you might also try a set up to keep them cool at night if you do not have airconditioning. A cool pad to sleep on for example
http://www.amazon.com/gp/product/B0009P42IC/103-8334819-2381433
A fan blowing in their direction might also work.
Before trying anything over the counter I'd have your vet check the worst one out to make sure it was going to be safe to give them something.
If the problem is kennel cough you might find some Robitussin will help read cautions for use here and dose info
http://www.petplace.com/drug-library/dextromethorphan-robitussin-dm/page1.aspx
But if its allergies Benadryl might help
http://www.petplace.com/drug-library/diphenhydramine-benadryl/page1.aspx
or one of the other antihistamines which your vet can talk to you about.
The vet also has good meds to help open bronchial tube areas.
My concern would be that you might 'hide' a serious issue with soft palate or nasal passages doing your dog harm with home meds. So I really suggest seeing your vet to make sure you know what exactly the problem is.
Hope this helps you!
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