If you think about what happened when you got this infection. You swallowed the supplement. Where did that then go to? Chances are you already had an infection which was suppressing your stomach acid and that prevented your stomach from destroying the bacteria in that supplement. This is so common, people fighting candida and numerous other infections. They rarely fight the correct thing and only when they use more general methods do they manage to beat what they are fighting. People fighting candida will use anti fungals until they are coming out of their ears but only when they use something which fights bacteria do they beat it.
From what you say you believe that you have Increased Digestive Permeability aka "Leaky Gut". This would fit the stomach infection because all Leaky Gut originates in the stomach and because of the way the vagus nerve functions it also affects later parts of the digestive system causing increased permeability in those other areas too. To live in the stomach, bacteria need to create a biofilm and it is the biofilm which causes the Increased Digestive Permeability and biofilms dramatically increase antimicrobial resistance.
I would not waste time with Oregano if I was you. It never really did anything for me. There are far more powerful Antibiotic substances than oregano.
IMO Ceylon Cinnamon bark essential oil would be considerably stronger. That can even destroy the majority of bacteria inside biofilms and works against the majority of microorganisms. You do need to be very careful with Cinnamon eo, you cannot just drink the stuff. It needs to be consumed either mixed with food or put inside capsules so that it can be swallowed. I would read up on it before using it to be safe. Some people do add Cinnamon eo to food without issue but drinking it can affect the mouth and throat so it should not be consumed neat.
I intend using it myself. I do want to emulsify it using an ultrasonic cleaner though so that it works over a larger area. That will prevent it from coalescing which would reduce the area that it works over. Reducing the biofilm would increase the effect. That takes enzymes and quorum sensing inhibitors. I am expecting some QSI in the next day or 2. BFB 1 and BFB 2.