If orodental causes of the halitosis
are ruled out, it will be important to rule out the next possible level of the GIT i.e oesophagus which you might either be having hertial hernia or gastrooesophageal reflux diseases. So transoesogastroduodenal fibroscopy/upper GI series has to be performed to rule out the hertial hernia.
In any case those need to be done and rule out any chronic condition like diabetes mellitus
, renal diseases, metabolic conditions including dyspepsia
To neutralize gastric acid
you can take sodium bicarbonate
salts or antacids and if you were in my consultation i will include a protein pump inhibitor ( even though you do not have epigastric pains).
I will also advise a surgical consultation or you see your doctor to rule out above and then decide to refer you to the surgeon.
Wish you a better health