Is there a way to add a custom field to the Inbound SMS activity type?
How about adding another activity status? Like "Unread" or "Read".

Depending on how skilled you are, maybe it is possible to have inbound sms set to "Unread" as default. Then your staff could change it to "completed" when they have read it.

If not, you could add the activity status "Read" and then change the status from "Completed" to "Read".

In both ways, you could have a report showing only the default activity status, and after they have changed the status, it diseappers from the report.