I've heard of this bug before. It's actually really amusing to me in the context of the story, but I'm sure it's frustrating to have to deal with.
Some people think this happens if you were a vampire when you installed Dawnguard. I can't confirm this, but it does fit your story. At this point, curing your vampirism and trying again still won't fix the issue.
Here are some things that will hopefully work.
PC Players - Console Commands
BEFORE talking to Harkon, open the console with ` (the key to the left of '1') and enter the following console commands...
setstage DLC1VQ02 40
setstage DLC1VQ02 180
Talk to Harkon again. He will say that you have forsaken his gift, but will offer it to you again. If you accept it now, you should receive the Vampire Lord power without dying from his bite.
Console Players (360/PS3)
Go behind a table before getting bitten. He won't reach you in time if done correctly, and the screen should fade to black after a short period of time. Following this, everything should continue normally.
Let us know if the above fixes solved the issue. Knowing what's successful will likely help people in the future.