It is possible to skip the King Bulblin 1 fight with a precise ending blow.
Lure a kargarok to the middle of the gate and hit it over the gate. Then, get close to the gate and use ending blow. This is precise, but if your position and angle were correct, you should barely clear the gate. After clearing the gate, roll around the trigger on the ground to enter Kakariko Village.
Sadly, the kargarok only has enough health to attempt this once. If you fail, you have to either fight King Bulblin 1 or reload the area to reset the kargarok's health.
This trick is made significantly easier with the use of moon boots, which allows us to gain significantly more height on the ending blow.
Lure the kargarok to the middle of the gate and hit it over towards the side, then perform a moon boots ending blow to clear the gate. Any of the first several frames will work for this, depending on position and angle. After clearing the gate, roll around the trigger on the ground and enter Kakariko Village.
If the kargarok lands far back, an early moon boots could put you into the King Bulblin 1 trigger. If this happens, wait a few more frames before force unequipping the iron boots, which will cause the ending blow to cover less distance.
If the kargarok gets stuck behind the gate, it will ignore you and fly away. But you can prevent it from doing so by using the boomerang to get it into position. This is more difficult than the normal method, but it saves time over waiting for the kargarok to notice you again.
Alternatively, you can bonk through the gate with Epona like with Ordon Gate Clip and Early Lanayru Province. The lock being in the way makes this clip a bit harder than those, though.
For Phase 2 on the bridge you can turn around and run along side him to hit him. This is done by dashing 3 times then turning around just before the crack, then avoid him and knock him off
By voiding out in the area before the King Bulblin 2 fight, it is possible to skip the fight altogether. This lets us get Zora Armor without the bow or bombs. To do so, perform the following steps:
King Bulblin 2 can be defeated without the bow by throwing bombs at him with precise timing. It takes three bombs to kill him (unless you've already beaten King Bulblin 1), and the timing is different for the third hit.
Orcastraw made a setup video that pause buffers the bomb pulls and throws.
If you defeat King Bulblin 1 before fighting King Bulblin 2, King Bulblin 2 will take only two hits instead of three.
Unlike in the SD versions, you cannot hit King Bulblin multiple times with a single quickspin. Instead, jump attack King Bulblin as soon as he starts his attack animation, then chain quickspins. After the fifth quickspin (including the one in the jumpspin) backhop to evade his attack. Repeat the process to finish him off.
If you deal too much damage without defeating King Bulblin, he will do a tornado attack with his axe, which wastes a bunch of time. But if you mess up in the first set of quickspins, you can avoid this by doing a few quickspins instead of a jumpspin and doing the normal second cycle after that.