Use Tianji Website monitoring for other docker containers?

I have self hosted docker server on VPS, I have there different open source projects like poste.io, authelia, Vaultwarden, Next coloud ....
I am ok to set this up for websites I have created, by adding the script as given when the record is created.
Is there some way I can use Traefik or docker to implement this for all the containers I would want?

As currently I can see only solution to find the main html and add it there with that in mind that I need to keep the file permanently stored.
This can introduce issues if the file gets changed by project and I update the container.
But also issue would be if the project does not have some base html, like I have on flask websites, then I would need to add it to all HTML files in project.

Am I missing something or the above is only way how to add the monitoring to other projects what are not managed by myself?
Thank for advice.

No repository file, test, or entry point is identified in the issue. Start by locating the existing website-monitoring setup and its script-based integration, then clarify how Docker or Traefik-managed containers should be monitored and what behavior would count as complete.