AFAIK, the relinking requirement only applies if you distribute the software to someone.
If you’re running Sidequest entirely on your own infrastructure to orchestrate jobs across your backend, you’re not distributing the software at all, you’re providing a service. The tight coupling does not itself trigger extra obligations. What matters legally is distribution, not architecture.
Edgecase is if you give your software to a customer to run on their own servers (self‑hosted deployment/docker image shipped to customer). In those cases, you would need to allow them to replace Sidequest.js (ie, not obfuscating it away).
Someone more knowledgeable can correct me, if I'm wrong
Layman here also, but I think you are correct about GPL and LGPL (this case), but not for AGPL which adds a requirement that: "... If your software can interact with users remotely through a computer network, you should also make sure that it provides a way for users to get its source"
https://www.gnu.org/licenses/gpl-faq.html#AGPLv3ServerAsUser