You are describing an infant with rapid weight gain and height velocity. I am happy you saw a peds endo, rarely we see infants overweight and are concerned, as they cannot walk yet their intake is controlled by the family and a dietary history is critical. Is she taking a bottle-if so what is in it? is she on solids? is she in daycare or at home. Ultimately the intake and output of energy are off, sometimes related to hormones or genetics.
So tests include thyroid and in an infant who has poor tone and other concerns I may test for Prader Willi.
There is alot of research on obesity and many folks are looking at factors like leptin or MCR-4 deficiency, factors that we are learning about that control our metabolism, these are rare indeed but so is an overweight 9 month old.
firstly, she needs to be seen by a dietician to assess her intake versus her true needs, next consider working with your peds endo, if she has testing done at the place in England often it is just a simple blood test that is sent from your doc's office for testing over to the lab in England. I use the UK for rare tests and they are very helpful and many folks that is all they do in their lab is investigate very specific questions such as this.
The fact that your husband and his family are overweight does bring up the nature versus nurture question, yes, she may have inherited a slower metabolism but also the food that is offered to her and portions and frequency of feeds may be oriented by an environment that promotes this.
We don't want to miss anything, Albright's was a keen thing to think about-this is pseudohypoparathyroidism-or where the body is resistant to several hormones like parathyroid hormone, thyroid, growth hormone, these kids are short usually, have a shortened 5th digit. It is rare but important to think about.
Her rapid height is probably driven by her weight, excessive body fat makes for insulin resistance and this leads to advancing height velocity due to effect of insulin on the gh receptors.
I would start with dietary involvement on a monthly or bimonthly basis with a goal of maintaining weight rather than ever thinking about weight loss.
Then stay with your endo and give it sometime. Be very very very truthful to yourself about what she is taking in dietary wise and how your family can perhaps make changes to help combat genetics.
Keep us updated.