So we are supposed to believe that the Potions Master would have no antidotes or tricks up his sleeve to combat snakebite?

No. We are supposed to believe that the Potions Master, DADA expert, and Head of Slytherin, who had been leading a double life for decades, devoted to bringing down a dark wizard with a notorious affinity for snakes, would have no tricks up his sleeve to combat snakebite. Snarl, I think, doesn't quite cut it. If I was Rowling, I'd call it a Snape-shaped plot hole, but I can't sink that low without lots of booze.