It's a time thing. Lee already had light sensitivity and mental impairment by the time they got to the morgue. Too much time already passed and the 'infection' spread when they got there.
Also, in my playthrough, Lee passed out before they cut his arm off. You don't have a healthy, middle aged man pass out from a bite unless the bite also did something to him.
So yeah, it wasn't he blood loss that did in armless Lee.