So while the antivenom will work fine the first time, the second time you might be immune to it because your body might make antibodies against the antibodies from a horse or bunny. This is one of the cases where the immune system just could not have expected that modern medicine would come up with creative solutions like pumping venom into a horse and then using its blood for ourselves. Which is fair so we can’t really be annoyed too much with our immune system in that case.