Inconsistent parenting causing daughter's weight gain

by healthy
(melbourne)

Parents and caregivers are the number one cause of childhood obesity. Take me for example.

I'm an educated healthy male who strives to be healthier every day. Since seperating from my wife 2 months ago, my 4 year old girl has gone from being a healthy 17.8kg to 20.4kg in that small amount of time.
Fair to say that increase would be justifiable over a 2 year period, but not in two months.
Here's the problem and I'm sure everyone can relate. My daughter rings me up whilst in my ex-wife's care and demands a donut, but mummy says no we're having yoghurt. Daughter screams and demands a donut. Mummy says no, yougurt.
Tantrum ensues and the line cuts out. I sms "good parenting" to my wife.
Later that day I take my daughter to a party and she announces to me that she had the donut and a lollypop, then ate chips and ice-cream at the movies. My daughter has learned very fast that screaming gets you what you want.
During the party, I tell her that after three nuggets, chips and fairy bread enough is enough.
I pull her from the at the beginning of a tantrum and gently, but firmly tell her that we will leave the party at once if she continues. I say no mean no and that's it.
After 30 seconds she accepts it and when she calms down I tell her that one day you'll understand and daddy loves you very much, but eating junk food regularly is unhealthy.
Here's the thing: People are so afraid that if we say no to our kids, we are bad parents. Or if a child starts having a tantrum, we give in to save face.
I was put in a situation where my daughter was about to erupt in front of about 100 people. Instead of giving in, I took her away from the situation and perservered. If my wife would have done the same for just a bit longer, she would have had the same outcome and it would have gotten easier every time.
We can pack healthy snacks to avoid fast food. Eat healthy with our kids too. Get off our behinds and excercise with out children.
There are no excuses and we are 100% to blame as I've identified today.
My question is how on earth do I stop my girl from her escalating weight gain when I only see here 1 week on/off. It's like climbing a mountain.
She put's on 1kg every week she's with her mum and then loses about 0.5kg with me. You can see the constant gain, even when I take her swimming, dancing, running. I cannot communicate with my ex and vise versa as we are both under intervention orders so I just can't simply pick up the phone and ask her what she's thinking.