Destroying "Fighting Spirit" when the equipped monster would be destroyed does not use the Chain.
"Fighting Spirit" is destroyed when the equipped monster would be sent to the GY by battle. If this effect is used, "Fighting Spirit" will not be on the field at the end of the Damage Step and could not be targeted by effects that activate at the end of the Damage Step.
If a "Hyper Hammerhead" destroyed by battle is equipped with "Fighting Spirit" and it is destroyed instead, "Hyper Hammerhead" remains on the field and its effect activates.