I am the developer of the mentioned monolog-viewer, thanks Carlton for recommending it!
It does allow you to filter by log level and you can either install it once and access log files from multiple domains via FTP or install it on the same domain as the application and access the files locally.
If you think a feature is missing, feel free to open a GitHub issue and I'll see what I can do.
I haven't tried it yet but https://github.com/Syonix/monolog-viewer seems like an attractive solution.
If you are willing to change the handler that you have configured Monolog to use you could consider pushing messages to a 3rd party. They would present the logs in a pretty format with good filtering functionality. It's likely that these services may charge you for the service though.