So if you were BP would you want this pulled down? If you were Twitter would you comply? 

Most brands need to protect their properties and profiles. Whenever possible, get the “official” moniker from the network. Also, purchase up some of the common variations of the businesses name, and if someone squats or jacks a profile or pretends they are you . . . go after them. 

But in this case. BP and Twitter should do nothing. If BP tries to pull this down, they’ll just come off looking like not only the largest polluters on the planet, but bullies who can’t take a joke. And if Twitter pulls it down, they’ll look like spineless dweebs who won’t stand up for the average social media geek.