I believe that you could have a leaky gut which is what could be causing the lymphatic problems, the skin problems, the energy problems, the hair loss problems and possibly other things.
That would lead to food allergy because undigested food molecules will be absorbed into your blood stream and your immune system will react to them.
There are different food allergy tests available and their reliability is not meant to be great. I found a simple skin prick test told me all I needed to know. They scratch the skin and put a drop of allergen onto it. Wait and if the skin becomes red and/or swollen if means that there are IgE antibodies to the food. However there are multiple anti body possibilities with food allergies . If you have a leaky gut, avoiding the food will make no difference. I came up with a way of lessening the symptoms but it might not work for anyone else and it is very difficult to do. For me it is worth it.
I don't think that a food allergy test will be of much benefit except to show the likelihood of a leaky gut.
I also believe that your stomach not healing is due to what I have said all along, a biofilm with a likely urease positive bacteria present which is neutralizing your stomach acid whenever you consume anything which stimulates acid production. Your burping sounds minor but there does seem to be something producing gas. Mine is much more pronounced but I have a very large amount of permeability and biofilm. I will explain how I know this later.
A Urea breath test [UBT] would confirm whether there is any urease positive bacteria present in your stomach. You have already had a Pylori anti body test which came back negative so a UBT should not detect pylori only other bacteria. The UBT can give false positive results for h. pylori because other urease bacteria can be present which can be interpreted as being h. pylori.
The UBT was the test that I asked my doctor for but she opted for the cheaper ELISA h. pylori anti body test.
There does seem to be a Urease anti body blood test available which might be cheaper than the UBT and should provide the information which would indicate whether a urease bacteria is present in your stomach.