I have not done this as you describe, but I don't know why it would be out of the question with enough work. If you have a method of exposing data from your LTI to the client side in some way you could add some JS to add an element to that line item. Maybe use custom data? Users - Canvas LMS REST API Documentation
Would it be neat to have? yep. Would it be way more work than one would expect? Also, likely, yep.
It is a good idea, but I think your time would be better served putting something together to send a message or an email for notifications. Conversations - Canvas LMS REST API Documentation
Canvas gives users/institutions quite a bit of control, but the onus is on you for maintenance so I am always hesitant to make broad changes to the actual functions of Canvas with JS, unless you have someone/a team that just builds/maintains your custom integrations to do upkeep and check patches I probably wouldn't recommend this route.
Hopefully this helps.