Blinkbox does not support AirPlay, so won't send to AppleTV. Would therefore not surprise me if it does not support any other type of connection to TV (didn't know you could connect IPad to tv in any other way).
So even if you do get other videos to play, Blinkbox videos may not play.
Blinkbox Faq says connection via HDMI cable is not guarenteed. So maybe that would work though it may not.
If you want to watch pay-per-view/rental movies on TV, have a look at AppleTV which will let you rent via iTunes, and ChromeCast (lot cheaper than AppleTV but is brand new so not sure what video rental services support it currently).
AppleTV and ChromeCast support Netflix, so for older movies that could be useful.