Once you have the custom report template uploaded to your Business Analyst Server instance, in the Flex API you would use the same method for creating the report from the custom report template as with any other report template that comes with the default installation. You can just use the Flex API SummaryReportsTask and the SummaryReportsParameter classes. In the SummaryReportsParameter class, you would specify the boundaries parameter to be the featureSet representing the trade area and specify the custom report template name in the reportOptions parameter.
If you are using custom data with your custom report template, then you would need to also specify the dataHierarchy parameter to the path of the custom dataset. You can use the GetCustomLayers task to get this path.