Also the task visibility and reassignment "rules" are followed for the WebUI and REST API but ignored (by design) for the JS API. So if you created a dashboard for the support group and used the JS API you could display the tasks to the support user and allow them to reassignment.
If you really want to use the REST API, tw_admins also don't have the visibility / reassignment rules applied to them, but this may cause your support group to have greater access to the BPM Environment than you want.