He just bites down at first.
His real game is provoking Edward. (Bella is unimportant so will become first lunch and then a statistic... after the fight.)
From the script:
JAMES
Sorry. But you really made it too easy.
....
So to make things more entertaining,
I'm going to make a little film of our
time together.
....
He will. His rage will make for
more interesting sport than his
feeble attempt to protect you.
Then Edward arrives.
Looking directly at Edward with a sinister grin -- he SINKS
HIS TEETH into Bella's hand! She SCREAMS in agony.
Edward extracting the venom.
After the fight only dr. Cullen, Edward and Alice are able to control themselves enough to be able to help. Dr. Cullen is busy stemming Bella's blood loss. Edward is the only one having a prayer of succeeding to suck out the tainted blood, and then stop drinking.
Abbreviated script:
BACK ON BELLA -- as Edward reaches her side. Dr. Cullen works
fast to assess her wounds, focusing on a massive bleed from her
leg. But clearly her hand hurts the most. She writhes in pain.
Dr. Cullen applies pressure to the gash on her thigh -
DR. CULLEN
Her femoral artery's been severed.
She's losing too much blood.
You have to make a choice. Either let the change happen -
EDWARD
No!
ALICE
It's going to happen. I've seen it -
DR. CULLEN
Alice, find me a tourniquet. A rag, a shirt.
Alice holds her nose and avoids the blood as she rips the
sleeve off her blouse and kneels to aid Dr. Cullen.
EDWARD
What's the other choice? You said -
DR. CULLEN (to Alice)
Tighter, above the knee -
DR. CULLEN
You can try to suck the venom out.
EDWARD
I -- won't be able to stop -
DR. CULLEN
Find the will to stop. But choose, she only has minutes left.