Where in sitecore do I configure what fields are displayed when I click this button? (Currently, it lists 3 of the 7 fields that are on the template, I'd like to add another to be editable).
It is one of Edit Frame buttons. When you open your control markup, you see something like this:
<sc:EditFrame ID="editFrame" runat="server" Buttons="/sitecore/content/Applications/WebEdit/Edit Frame Buttons/Default">
...
</sc:EditFrame>
It differs for MVC, but is similar. You need to get Buttons attribute value.
When you open /sitecore/content/Applications/WebEdit/Edit Frame Buttons/Default path in core database you see list of children. It is your buttons. For your case it should be only one button: Edit Item. This item has Fields field where using pipe as delimiter is list of fields that are available for editing.