Betaine-HCl is often recommended to increase stomach acid and thus digestion. However, I found this comment by Art Ayers in which he refutes that it would work. Could it actually be counter-productive? What about a supplement that not only contains betain, but also enzymes like pancreatin, papain and bromelain?
At the link that you cite, they were advocating adding a salt, betaine-HCl, to increase stomach acid. This is silly. The betaine is a weak base and when it is neutralized with HCl, you get the salt. That just means that it acts like a buffer to maintain a particular pH, it is not going to acidify any more than adding NaCl. Betaine is, however, an interesting compound that may interfere with heparin-based signaling.
The point here is that alteration of stomach acid levels is not caused by the contents of a single meal and indicates more profound problems, such as the most likely, Hp infection.
I don't buy the impact of stomach acid levels on protein digestion and some consequence of proteins slipping through. That isn't consistent with how digestion works. The peptic enzyme hydrolyze proteins differently than the subsequent pancreatic proteases. This is interesting with respect to antimicrobial peptide production, but I don't think that there is going to be a systematic impact on immunity. Those proteins and large peptides do not get across an intact intestine. Leaky gut is required. COX inhibitors would be more likely problems.
Vinegar and lemon juice are not going to acidify the stomach. They contain weak acids, acetic and citric acids, that will actually act as buffers and neutralize stomach acid. That doesn't mean that they are not useful, because they are very helpful in dissolving biofilms.