Hello Seven. You need to show the time difference between the snakebite (an instant only) and his lasting fear of snakes. Since his fear is in past tense (was), the event of the snakebite should be shown to be earlier, Having been bitten...
Here are examples of the participle alone as the start of a chain of events (not a lasting condition, like the fear of snakes).
Wakened by a noise in the street, Harry jumped out of bed and ran to the window.
Damaged in the storm, the house needed expensive repairs.
Lost in the snow, the explorer tried to use his compass to get back to civilization.