Ok, here goes, hope this helps.

We have a custom property editor called "CategoryPicker", in it we have a CategoryPicker.controller.js containing this:

CategoryPickerResource.getChildren(rootNode, "<pageType>").then(function (response) {
    $scope.documents = response.data;
}, function () {
    notificationsService.error("Error", "Error loading documents");
});

• the "rootNode" is a prevalue that we set up when using the datatype, it's pointing to the parent node under which the categories are children.
• the "pageType" is the document type we want to filter the list of children by

This then uses the CategoryPicker.resources.js file to call the API, this code is:

getChildren: function (rootNode, documentTypeAlias) {
        return $http.get("backoffice/Plugins/CategoryPickerApi/GetChildDocuments/?rootNode=" + rootNode + "&documentTypeAlias=" + documentTypeAlias);
    }

This then calls our "CategoryPickerApiController.cs" which is contained in our "App_Code" folder. This contains:

public IEnumerable<DocumentInfo> GetChildDocuments(int rootNode, string documentTypeAlias)
    {
        var contentService = ApplicationContext.Services.ContentService;

        return contentService.GetChildren(rootNode).Where(x => x.ContentType.Alias.Equals(documentTypeAlias)).Select(x => new DocumentInfo { Id = x.Id.ToString(CultureInfo.InvariantCulture), Name = x.Name });
        //return contentService.GetChildren(rootNode)
        //    .Select(x => new DocumentInfo { Id = x.Id.ToString(CultureInfo.InvariantCulture), Name = x.Name});
    }

You can see in the resource file that the response.date was set to the $scope.documents, that populates the dropdown in our editor.html:

<div ng-controller="CategoryPicker.Controller">
<select name="CPDropdownList"
        class="umb-editor umb-dropdown"
        ng-model="model.value"
        ng-options="d.Id as d.Name for d in documents" />

Obviously these aren't full code files, but the main bits from each, I'm not writing the whole thing for you :-).

Let me know how you get on.