Sorry, but the only tests for the LES are barium swallow and manometry. Just looking at the sphincter will only tell if you have a hernia or visible damage. Generally GI docs like their patients asleep and not gagging for endoscopy, so you can't swallow, and the doc can't see if the LES is functioning.
Vinegar or any other acid lowers the pH of the stomach which signals it to stop making acid. There shouldn't be any need to take a specific kind of acid. Lemonade might work.
All the pH raising waters, Tums and baking soda and such reduce the acidity of the stomach, which gives temporary relief from the burning feeling. Mineral water at least tastes good to me and works fine for the short term.
The formulations for most PPIs have 24-hour release capsuls now. Unfortunately these only treat acid, as do the home remedies. The only treatment for a non-functioning LES is one of the surgeries.
If you decide to try a home remedy on a long-term basis you should at least consult your doctor first.