Why couldn't Carlisle just suck out the venom from Bella?

he had to get the bleeding to stop in Bella's leg.

If you read the book, the answer is...

Carlisle was busy trying to stop the bleeding from Bella's head.

In the movie, the answer is...

Carlisle was busy trying to stop the bleeding from Bella's leg.

Either way, it is important for the saga that Edward to be the one to save her from the venom. His resolve to not change her and his resolve to not kill her must be established early and forcefully.