Wouldn't most low carb diets involve lots of animal flesh which is harder to digest, so might contribute to OP's problems?
No, low carb diets DON'T mean lots of "animal flesh" or meat, I presume you mean maZebraltov. You can be a vegetarian, and still eat low carb. But it DOES mean upping the amount of healthy fats you eat, as they become your main source of energy instead of sugar/carbs.
And as OP is not overweight, she would need to eat plenty of fat to keep calorie intake up.
OP, I used to suffer from dreadful reflux problems, and adopted a low carb way of eating two years ago. The reflux disappeared almost instantly, to my great relief. However, I was also overweight (same height as you, but 11 stone 3) and the weight very slowly came down by 24 lbs. My weight is now stable at 9 stone 7, and while I could do with a few more pounds gone, I'm happy to stay at my current weight.
You could check out the low carb diet boards for ideas or Google Diet Doctor, which is the low-carbers bible - full of lovely food ideas and interesting info.
Far better to change the way you eat, than rely on medication such as PPIs, which, while having a short-term effect, can create big problems further down the line. Such as, "rebound acid", which is worse than your original symptoms, and with long term use, the risk of stomach polyps, and non-absorption ova vitamins and minerals - pernicious anaemia for example.
I'm 71, and didn't think I'd be able to adopt this way of eating - I have cut out bread, potatoes, rice, and other starchy foods, and I don't miss them. The upside is you will feel far more energised, your skin might improve, and simply getting rid of the reflux is good news for your oesophagus!