This is certainly an uncomfortable constellation of symptoms, as well as potentially concerning for an underlying medical condition. While it is not possible to make a diagnosis without having an actual office visit, taking a more thorough history and performing a physical exam, there are several possible causes for your bothersome symptoms and would urge you to further discuss with your primary care doctor
1) Congestive heart failure: it is possible that the your symptoms are due to mild or moderate heart failure. When the heart is in failure, it is less efficient at pumping blood forward, so blood becomes 'congested' in your body's blood vessels. This can cause dyspnea from pulmonary edema (fluid backing up into the lungs), bloating or increased abdominal distention from blood/fluid backing up into the gut wall, liver and other abdominal organs, and weight gain from backing up of fluid into the skin, causing swelling or edema (most commonly in the ankles and legs).
2) Diabetes: there is a strong association between PCOS and diabetes. potentially the bloating is due to gastroparesis (delayed emptying of the stomach) in the setting of uncontrolled diabetes, the bloating/distention itself could be causing the shortness of breath if your abdomen is becoming distended enough to exert pressure on your diaphragm, thus making it hard to get in a good, deep breath.
3) Dietary intolerance: it is possible that even though you are cutting out common 'gas-inducing' foods, you still may have an intolerance to a type of food or a food group (eg: dairy) that is causing your symptoms. A food journal may be helpful in sorting this out.
I hope you find this information helpful, but please continue to work with your primary care doctor on these issues.