What is important on this to understand is that even if your business is primarily a processor, it is a controller when it concerns a FB Page. In order to not get dragged down with the complexity, I recommend focus on decisions you make in the role as controller on the FB Page.
For example. You business is not a controller on FB when it concerns the FB user account. This is FB. But if their FB name (and more) is used with the intention of e.g. creating an offer and/or marketing, your business is a controller. Even the pop-up Chat dialog asking if the visitor to your Page would like more info on the services, are you a controller.
What this means is that you must have a link to your organisations Privacy Notice from your FB Page or have the Notice in FB. Think about whether your business really needs a FB Page? When was the last time it was updated? If it was an age ago, and is dormant, maybe it's better to get rid of it.