Keep in mind that optical discs burned through a consumer grade drive will ultimately degrade over time (faster if exposed to heat and/or sunlight) and you’ll need to periodically check the data integrity of these discs, on top of keeping a backup somewhere else in case there is data loss due to degradation.
Honestly, you’re better off buying more storage and have a good backup system than to waste money on BD-R discs which will end being most costly and more work per GB.
If you still plan going that route, you could use a prefix / letter to indicate which type of media it is (ie: MOV for movie, TV for series), a single letter for the main genre (ie: A for action, C for comedy, etc) and use a unique number to identify each disc. If it’s a TV show or a trilogy, etc, you could add a suffix (ie: .01, .02, etc) so that you can regroup and find them all.
Personally I use the imdb ID (ie: tt1190634) in the filename, as that will be easy to find no matter the language used.
Could you power cycle both the modem and router?