My kids ate a lot more than that, but then they were and still are big eaters and big boys (Tall not fat!)
I would think that you need to be giving her snacks between the meals rather than giving her bigger meals if that makes her sick. My kids never went more than 2 hours without something! (still don't) Try her with some finger foods like toast, cheese, fruit, pasta and biscuits or cheese straws or breadsticks or something in between her meals.
My elder son used to projectile vomit - lovely - and was always sick when he had a cold or a temperateure, so often lost weight or didn't gain any when he was unwell, but you need to maximise the opportunities for her to intake calories during the day. That might help her gain a bit, and alsays make sure that she has the highest calorie version of anything - she needs her fats, which is probably harder if you are avoiding dairy.