No, Bingle Bangbang's ability will not trigger when it's being attacked.
According to Keyforge's Rulebook (May/2019 version), under Using Creatures, the rule Fight reads:
[...] After the fight resolves, if the creature that is being used to fight survives, all “Fight:” abilities the creature has, if it has any, resolve.
While, under Glossary, the keyword Before is explained:
If the word “before” is used in an ability (for example, “Before Reap:” or “Before Fight:”), that ability resolves before resolving the game effect of the reap or fight (but after the card exhausts, if exhausting is required to use the card).
As you can see, the rules for Fight clearly state that if the creature being used to fight survives, all “Fight:” abilities the creature has, if it has any, resolve. Since the ability of Bingle Bangbang is a Before Fight trigger, would trigger before the fight resolved, if it is being used to fight.
When being attacked, creatures' Fight ability won't trigger.