Syncing inventory between Amazon inventory and Infusionsoft's inventory

I am trying to help a new Infusionsoft user to sync their inventory inside Infusionsoft and Amazon. We are going to use Ship Station. They are also willing to look at using WooCommerce if it would achieve this.

The current setup is to use Infusionsoft shopping cart/order forms to then connect to Ship Station. Ship station will connect to Amazon FBA for some products, but not for all.

They ultimately want a purchase in Amazon to update the Infusionsoft inventory and vice versa.

Any help would be greatly appreciated!