Hi Nimrod,
Welcome to community.
I go through the post carefully and understand you want to keep the subdocument linking with the master document after you change the name/title of the subdocument. (If I misunderstand anything, feel free to correct me)
When I test from my side, I can reproduce the same behavior that after I change a subdocument's name, the master document will show that the subdocument is missing, so the subdocument stops linking with the master document.
Unfortunately, I'm afraid it's the expected behavior and there is no out of box way to do what you described. Really sorry for the inconvenience.
Given this, I sincerely recommend you send feedback via File>Feedback within Word. Feedback submit by this way directly goes to the related team. This is the best way to let developers know your requirements and suggestions.
In the meantime, I recommend you also refer to Doug's suggestions.
Thanks for your understanding. Have a nice day and stay safe.
Best Regards,
Tina