I haven't done much with writing LTIs, so I may be way off here. Keep that in mind as you read it.
I don't remember seeing LTIs needing this level of integration so it makes me wonder if you're making it harder than it needs to be.
It also sounds like whoever is giving you access to their system is giving you a lot of access, which is probably why I was getting confused about which side you're coming from. Normally LTI's don't need that level of access to function, they can operate within each course without having to rely on information from other courses. Typically you do not need the internal IDs from Canvas unless you need a high level of integration with information that is only available through the API and I'm not seeing that here (at least not yet).
How did you do the original mapping? That is, how did you decide that assignment X in Canvas needed content Y from the LTI in the first place?
I remember using one LTI many years ago that had a single URL for all of the assignments, but I had to go into the system and configure the assignment before each one. It was a classroom response system that could only have one response session open at the time (all I was paying for).
Since then, when I have setup LTIs in my courses, they vendor provides a mapping for me. That is, I create the content in their system and import it into Canvas. Either that or I go into Canvas and link to the content manually by creating external tools with query parameters that link to the LTI content like the pattern I showed earlier. Then, when it's copied, it doesn't matter because it links to the same content on the LTI side and the LTI vendor doesn't need access to Canvas API directly (which is good since some institutions really make you crack down on that).
If they gave you the keys to the kingdom, it may be that you could go into their courses and modify the external tool links to add a query parameter and then do that now before they've changed assignment names. Then when they copy it in the future, it remains the same. That is not something I would be comfortable with an LTI vendor doing in my course, though, so you would want to coordinate with the local Canvas admin.
Some other people who have done more with LTI may want to chime in. Some of them monitor the main question section, but there are many who hang out in the Developer Group.
I haven't quite figured out the new community software yet, but it looks like if you go to Group Hub and choose Developer Group that you can post questions there. In the old community, you could share content between groups and I just don't know if that's possible now. I'm not sure if this is the easiest way, but once you get to to the Developer Group, under "Content" on the right you need to click on Forums, and then it will pop up with an "Ask a question" button.
Also, I think I saw someone move a thread the other day, so I'll tag @Stef_retired and get her take on what the best way to get the Developer's Group involved is.