Oh, I got confused when you said 'TV Tuner'. The correct terminology would be video capture card. In the setup you described, the STB is the tuner; your card can only capture video from the STB outputs as I said.
An IR blaster to change channels on your STB should be fine. But you would need pretty long cables from your STB to your other room, both for the IR blaster and the composite cables. This will degrade video quality for the composite connection. Don't forget you need another set of cables for audio as well. The other thing I hope you know is that you can't watch one channel on your TV and a different channel on your PC simultaneously.
Tatasky has multi-STB options. It would be much better to get a second STB.