Gas build up in the stomach can be deadly It's called bloat: WVVH Emergency
A gas filled stomach can also lift the spleen and it can end up wrapped around the intestines.
I have a golden retriever, named HaHa with the potential for this this problem.
Anyway....and as far as the burping and air in this tummy - one of the solutions I came across was to feed him much smaller meals. It really helped with the really bad burping issues, although he still burps a bit...which is a Good thing...that just means that the gas isn't getting trapped in his stomach.
He still gets the same amount of dog food during the day, as a dog without issue that might get fed twice a day...he just gets more meals with small portions, I'm feeding him 4 to 5 times a day.
I usually add a little water to his dry kibble and let it get semi soft...but not soggy and at least one of his meals per day is dry kibble served as a treat, with me tossing a few at a time and let them scatter on the floor. He hunts them down and actually chew them before swallowing. He likes the crunchy, but if it was in a bowl he would just inhale it rather than chew it.
Also, upon occasion, if HaHa is looking a little distressed from gas, I give him a Gas X (simethicone) - it seems to do a good job of helping him burp. Also Gas X is really inexpensive... at our local dollar store I get 50 pills for like $3 or $4 dollars.
By distressed, I see that he's pacing a little and swallowing a bit after eating. I'll give him a Gas X pill and if he burps within 5 or 10 minutes, and looks like he's ok, then I relax...but if the day ever comes when he's not looking 100 percent after getting a Gas X pill, he'll be at the vet's office...as bloat could have happened and a pill isn't going to do anything for it.
Also, I did a lot of reading about colostrum for another dog I have that has some allergy issues. I was impressed enough by what I read that I not only did I give it to Jaya, but HaHa gets some too each day.
Colostrum has been studied by real universities and agricultural schools and it has a lot of benefits with the digestive system. It can repair thin walls of a stomach or intestinal track with keeps toxins from those areas from leaking through and into the body. It adds various elements into the system that are good for the body. For dogs with allergies, like my dog Jaya, sometimes it helps ease allergy issues.
I'm not a vet or anything so I don't know if any of the above will help your dog. The Gas X is not expensive, but the colostrum is a little more money. I bought powdered kind...but it's sold in capsules too. Make sure if it was heat treated it was done with low heat other wise all the good stuff in it is destroyed. I bought mine at Amazon, 1 pound of powder in a half gallon sized container for around $50.00...but it last a long time. I use a tablespoon a day and that is for both dogs. (1/2 tablespoon each). This is the kind I bought but there are other brands and such out there...some people like capsules as they are easy to give...also because it's kind of like a milk taste most dogs won't spit it out. Jaya and HaHa both will like any powder left from their dog bowls.
Amazon.com: Colostrum Bovine 16oz Powder #1 Best Value on Amazon 50% DISCOUNT TODAY! 100% Whole Nothing Added, Collected 1st Milking Only, Maximum Biological Activity, Contains Natural Occurring Probiotics, High Ig, Ld, Plus Lactoferrin, Nutritional Immune Support Supplement for Pets, Dogs, Kids, Adults, Athletes, GUARANTEED: Health & Personal Care Amazon.com: Colostrum Bovine 16oz Powder #1 Best Value on Amazon 50% DISCOUNT TODAY! 100% Whole Nothing Added, Collected 1st Milking Only, Maximum Biological Activity, Contains Natural Occurring Probiotics, High Ig, Ld, Plus Lactoferrin, Nutritional Immune Support Supplement for Pets, Dogs, Kids, Adults, Athletes, GUARANTEED: Health & Personal Care
So... you might try:
~Smaller but more frequent meals
~Gas-x when you see swallowing, or pacing...to help him burp and ease the gas pains in his belly.
~a long term approach with colostrum, I think it takes like a month or so for any benefits of it to show up. But again, I've read that it does a lot of good for the general health of a dog (or humans for that matter).
Anyway, your dog's issues could be something completely different from bloat. Check with your vet again, but from what I read about different digestive issues, I don't think any of the above things will do any harm.