LivingHurtEvent only triggers when a living is being attacked, where you can recalculate the corresponding damage when the attacker is holding your sword.
If you want to change sword texture based on the time, check out ItemModelProperties, the provider has world as one of its parameter which you can also apply the game time logic there as well.
You can use LivingHurtEvent and check if the attacker (DamageSource#getTrueSource, it can be null) is holding the sword, then modify the damage according to the time (World#getDayTime or something) there.
Or you can override Item#getAttributeModifers and apply the calculations there.