Hi Manson.
We have the functionality in the IC manager Business Role Email Inbox to route the emails to different organizations using RouteTo button in the Inbox search result page.
You can check that functionality and implement it in your business role i.e. provide a button RouteTo in the Inbox result and implement the logic same as standard.
For more info check the view ‘CRMCMP_EMWB/EMailWBSR’ of component ‘CRMCMP_EMWB’ ..
Thanks
Kumar