So whenever the card is held to the reader a script starts and plays the media and shuts down the TV after the episode has run.

It's easy enough to measure the accumulated script runtime and disable the reader once a daily allowance has been reached. Though I am not a fan of an allowance like that.

Nevertheless, tracking the script runtime is easy enough. It's a family TV, but a private TV can easily be tracked via power consumption. Then you can track any usage and help managing consumption.

Do you have episode run time just baked into each script or does it track via some other HA event to know when to turn off?

As I only play a single media file every time the script waits for the player going from "playing" to "stopped".