I think attempting to kick any dog that comes near your children in the park is an excellent and proportionate response and will really lead by example when it comes to showing your children how to behave safely around animals.
And yes I am being sarcastic!
I take my toddler and dog to Mayow park regularly and on the whole I think dog owners are responsible and there isn't a problem with uncontrolled dogs. It's a shame you've had a bad experience and unfortunately there is always the odd exception to the rule, it's a shame you've encountered them and I do sympathise if any children were actually bitten. Unless there is a very real and imminent danger though it is best not to run away from a dog as it may think it's a game of chase.
I appreciate it's difficult if she already has a fear but the best thing you can do is to teach your daughter not to be scared of dogs and to understand how to interact with them safely. Like it or not, they are here to stay so it is best to learn to coexist with them. She may even find some of them are quite nice!
Children "running around scared, screaming and with food" is not going to help matters as they tend to feed off our energy and may well think it's a great game. Or the dog itself may be scared. As the adult, if you encourage the children to remain calm and stand up and put yourself between the dog and the children a stern "no" or "off" may well do the trick, unless it really has approached the group with the sole intention of biting one of the children, which thankfully is fairly unlikely.
The picnic thing is tricky, usually a problem with young dogs and food motivated breeds like labradors. My understanding of Mayow is that generally dogs are supposed to be walked roughly around the edge of the park, which most people adhere to, particularly as the weather gets warmer and there are more other park users and less dog walkers about. I would suggest having your picnic near the middle of the park to minimise the risk of picnic rifling.