Found it. Two places need updating:
I added Jellyfin and Plex to the list of Excluded Applications, then added 127.0.0.1, ::1, this machine's local LAN IP address, and my public IP address to the list of Excluded IPs so that when I'm streaming to my own devices, either while at home or remotely, ESET won't try to scan my own traffic.
It'll be slightly less secure in the event that some program on this machine goes rogue and starts running a web server, but this is preferable to my SSD filling up all the time.