I am 5'3" and have been overweight must of my adult life. A couple of times I have slipped into technically obese and taken action to get healthier each time.
I weighed more than 10 1/2 stone when I left school at 18 but was very fit and toned and wore size 12 clothes, back when size 12 really was a size 12 . My most recent standard weight over the past ten years has been 11 4. This gave me a BMI of 29.
I've lost weight since my second child was born 20+mo ago. I have been as low as 9 stone through illness and I was shocked to see quite how much fat was still left! I have put on a bit of weight, prob half a stone.
I think we have all got far, far too used to seeing overweight as normal. Clothes sizes are not a reliable guide and they have clearly been altered to be more flattering in most British shops. It was very noticeable when I was in hospital that not a singe doctor was even slightly overweight.
I think you are probably not as unhealthy as most people your weight are (I've been pretty fit and heavy too) but that you probably do still need to lose weight. I'd throw in back in their court and ask them what they propose.