If a mass extinction is happening, climate change would not have had much time to factor into it. Most of the species loss has so far has had little to do with pumping carbon into the atmosphere. Humans as a species have ravaged the Earth in many other ways. Fishing the waters with factory trawlers, clearing forests for wood and palm oil plantations, carrying strange flora and fauna in the bilge of ships from port to port—all these things, and more, have contributed.
Considering the rate at which we are affecting the environment and the rate of our population growth, it would be difficult to argue that all of these current trajectories are stable. However, IMO even the eruption of a supervolcano like the Yellowstone Caldera probably wouldn't wipe us off of the face of the earth completely.
If I had to venture a guess, I would bet that the human population level will fall, perhaps by half within the next couple of hundred years. However, I don't expect that we will go completely extinct before we manage to acheive some sort of significant biotechnological transformation. That said, I wouldn't be surprised if this biotechnological transformation is what ultimately dooms us. By engineering ourselves, we will likely make ourselves less adaptive.