I would cut waaaaay back on the Kong stuffing. Of course she prefers it to her dog food--it's like offering a child a choice between a cookie or an apple. The stuffing has all kinds of yummy ingredients that make it taste good, ingredients that are NOT in her healthy dog food!
First, I'd never give her anything if she barks for it. This is teaching her that barking is the way to get things. As Geno said, make her do something and give the Kong as a reward.
Second, I'd hold off on the Kong brand stuffing. There are a million things you can stuff a Kong with that are much healthier while still being tasty. Check out the Kong website for some ideas
http://www.kongcompany.com/tips-and-expert-advice/recipes One of my favorite things is to use the water from a can of tuna to soften some of their kibble and then stuff the Kong with that. Or use plain yogurt mixed with kibble. You can use a different brand of kibble than what she eats for mealtimes so it seems like a treat.
Third, try freezing the Kong so it takes her longer to empty it.
Fourth, only give her one a day.
Plus what Geno said--make sure she's calm before you give it to her.