If you're taking 13 betaine hcl before burning, wow. I don't know why you say that's not an option because it's probably going to help you a lot.
Salt is good as it will supply chloride. Lemons might be really good as would other acidic fruits.
I posted an article about H.P. some time back. I'm not sure about chicken or egg, but H.P. presence can mean too high or too low stomach acid.
So I'd start with standard herbals like mastic gum and see if that helps.
Other bugs, infections, possibly protozoa could be in play.
google agni enhancers in aurveda: they include salt, ginger, lemon. You can take a mix of these three before each meal.
Beets supply betaine and can possibly help.
Low adrenal function is linked with low hcl acid, but you're back in a chicken egg scenario again not knowing which to tackle first. Just throw everything at it and see what sticks. If your stomach is out, you absolutely need to address it first since everything downstream from there will suffer.
Standard medical uses triple or quadruple Antibiotic therapy for H.P.
I'd hit up turmeric and Iodine as well to try to kill bugs in the stomach.
Vinegar is a good idea if you can tolerate it.
You should probably try gluten free if you're not already, as a lot of H.P. people say it's necessary, and those types usually do ok on paleo.
dark bags under eyes, = parasites
Iíve had h pylori tests that always seem to come back negative ... The only accurate test is endoscopic biopsy.