I honestly have to say as badly as they explored B'Elanna being half Klingon in Voyager, they may as well not have made her her Klingon at all. She could have been a human with a bad temper for all the good it did her and her story. Not much would have been changed by it. The two truly Klingon stories we got would have gone already, which always felt really forced and the story about B'Elanna having problems growing up could have been changed to her being a fat kid of something.
I'm not saying it wouldn't have been more interesting diversifying Voyager's crew a bit more then it was, but what I am saying is there was no advantage to having B'Elanna be a half Klingon and even if they would have written more on that angel, I don't think Dawson could have played off it better.
i think they explored it fine. she hated that part of heself and tried to ignore it, having a lot of episodes about her klingon-ness wouldn't have made sense