I'd check all the availiable non-controversial options first :) before delving into the more potentially risky ones.
I didn't want to list all my solutions earlier, as acid reflux is different than heartburn, but it appears much of what helped the reflux in my 2nd pregnancy is also suggested for heartburn.
-smaller more frequent meals
-staying upright after eating (don't eat right before nap or bedtime)
-sleep with your head higher than your stomach - for me, for reflux, some nights this meant sleeping almost sitting straight up, laying down just made things so uncomfortable
-glass of milk, (we don't drink milk, but I was almost willing to during the worst reflux days!), handful of almonds, bowl of yogurt (I like mine frozen, and sweet ... so I might as well have icecream ...)
-avoiding spicy or greasy food. Greasy was easy to avoid, spicy was much harder, we like garlic, onions, indian food, mexican food ....
-avoiding the foods that cause it, even if they aren't 'typical'. Everyone was telling me what to avoid and how to 'fix' things,but I just had to keep track for myself which things were ok, and which things seemed to make it worse, they weren't all logical. Coffee was fine :), some other normally 'ok' things weren't, and while the acid reflux lasted the entire 2nd trimester, the specific triggers seemed to change a bit along the way.
-eating slowly, sitting at a table, not slouching on the couch :)
I never did get around to buying the Papaya Enzyme last time, but will immediately if the reflux starts up again this time. There are some concerns about eating under ripened fresh papayas while pregnant, but I don't even thing we could buy them in our little city's grocery stores ... our available produce is not terribly exotic or tropical. I'd just get the bottle of tablets from the healthfood/natural products store around the corner.
Any of those options seem a lot more straight forward that either OTC or prescription meds, or taking essential oils internally, and there's no real risk at all to most of them (until you get to the potential for unripe papaya to possibly cause contractions).