Competitive external price is less than yours

We sell a product that is sold by a small number of vendors on Amazon, as well as many others online and in store. By agreement with the manufacturer, we must sell this product at $3 higher on Amazon. Because we offer free shipping on Amazon, this is actually a better deal than someone buying it from our website, where they also pay shipping. Sometimes, but not always, we get the message from Amazon saying that we are not selling the product at a competitive price. The better price online comes from our online shop, and other online retailers, and the manufacturer themselves online. Sometimes Amazon will offer a $3 coupon, so our product price on Amazon is better. Sometimes they suppress our Amazon listing. Sometimes they ignore it. What is the best way to handle this? I understand that Amazon wants people to shop on Amazon, but the manufacturer is not going to change their pricing. And Amazon doesn’t seem to understand that a customer actually does receive a better price on Amazon because of the free shipping.