Sorry the USB port on a TC cannot be used this way.. it is to attach a hard disk or a printer.. it will NOT give access to the internal disk.
I have tried to create a new partition in the Time Capsule "Data" folder using Disk Utility > New Image
Tried different formats (exFAT, FAT32) but the TV just does not see the volume.
Nor can you do this.. the internal disk of the TC is formatted HFS+ for Time Machine backups.. that is the only use it was designed for..
How can I make this work?
Buy a USB external drive.. they are cheap ..
Smart TV needs DLNA functionality to use video files on network resource.. TC has no media extensions.. not now .. not ever. Apple have restricted their systems to using iTunes.. and you must run iTunes on a computer somewhere in the network.
You can plug any proper Media Player into the TV and use it to play videos stored on the TC.. eg a WD live TV is a proper media player.. A PS3 or PS4 or any device that runs Plex can do it.. but it has to be intelligent.. ie be able to pick up a file in MPEG format (whatever??) and stream it.
People have used ipad and using VLC or some media player software, then used airplay to TV.. in the end this is just too complicated, as well as depending on wireless connections which are prone to slow video streaming.
An Apple TV is a streaming device.. it is NOT a media player.
You need to rethink your whole approach.