SSIS data flow to update source table rows after copying to destination
With T-SQL there's an output inserted clause which would be really helpful in this scenario.  

I'm not aware of a way to do that with SSIS, so your alternate approach of using a second data flow after your first (you can connect the second data flow to the first so your dependency will be honored), querying the web database, and updating the internal one as appropriate will work as long as your workload can tolerate the additional latency involved in waiting to update the SSIS_TRANSFERRED column.