There’s a usage (that I’ve heard mostly in Northern Ireland and Scotland) of putting “but” on the end of a sentence to mean “though”. E.g. “I don’t like her. She’s all right looking, but.”
The other theory is that the script has the “but” trailing off. “I don’t mind looking at TV but…” implying there’s something else to be said, that isn’t being said. Perhaps the actor has delivered the line badly.
Never heard it deliberately used at the end of a sentence. There are dialects that use strange words at the end of sentences like that - inner London accents use still, as in “I don’t mind watching TV still” and northern accents use like, as in “I don’t mind watching TV like” - but not but. You sure they’re not just trailing off halfway through the sentence?
I’ve heard it in Australian soaps broadcast in Britain. I’ve been known to use it myself for (hopefully) comic effect. The Australian usage definitely doesn’t trail off, “but” brings a sentence to a close as an intensifier. Racking my brains for an example but.
That’s how I use it too, but I’m American. The OED describes this as an Aussie/Scottish thing, so it may be different (though probably related).
It’s usually when I would otherwise repeat myself. “I just don’t like her. She’s nice enough, but… (I just don’t like her).” Or as above, when what would follow the “but” is immediately obvious.
OP might find this interesting: In German, sometimes they end their sentences with oder (or) instead of illustrating another option. Donno why; for me it’s kind of a trailing off thing like ‘but.’ For example:
Are you ready to go or?
Do you want to go to the movies or?