Bruno might have got him into F1 quicker than without using his name, but his lower formulae jobs did enough to merit his chance in F1. Bruno's main problem isn't his name, it's his success in lower formulae and his debut F1 season.
Now he's in F1, the name would have been a burden he had been a success - in 2010 it's just been a mention every now and then when he's had interviews. Had he driven some stonking races, he'd have had half the paddock around him asking him if being was God's nephew was a hindrance to his inevitable title.
As it is, he's been very average at best, so no one really cares (other than us fans wondering how the @#$%& Yamamoto is now as fast as him), so he's not had any real pressure from inside the F1 paddock. I don't know how things are with the Brazilian media, but from the rest of the media there's been very little acknowledgement. The only BBC interview since early in the season was recently when they happened to be talking to Karun outside HRT, and afterwards hijacked Bruno who was wandering around outside looking bored.
His problem isn't his name - it's showing he's got a quarter of the talent he appeared to show before F1.