The diagnostics from your living room device on 5/12 shows the following:
Filesystem Size Used Avail Use% Mounted on
/dev/fuse 5.2G 3.9G 1.2G 76% /storage/emulated
The OS is showing 1.2GB available. On FireOS if we try to use all this space, warnings start appearing on the screen and the OS will start killing apps. So we have to reserve atleast 750MB for the OS itself, which leaves about 350MB for your live TV buffer.
If you're watching OTA, its mpeg2 using about 18mbps (= 2.2MB/s). 350MB / 2.2MB/s = ~2.65 minutes
So yes, you only have a ~3 minute buffer before the player has to unpause.