You can 1 cycle this phase and skip the bomb fish coming out simply by quickspinning or jump attacking the eyeball. After this you want to do Swim With Water Bombs but DO NOT hold a direction, this will make you float up higher during the cutscene when morpheel comes out of the ground. If you don't float up it is because you starting swimming in a direction with the water bomb.
It it possible to regrab morpheel after hitting her four times instantly, this is useful when you don't have zora armour. To do this you need to get on her back with the Iron boots on, this is so when you force un-equip the Iron boots (by equiping an item over them) you will instantly fall off her without waiting for her to shake you off. You can then clawshot instantly to get back on, or equip boots and clawshot to get back on, although putting the boots on again is harder. It may seem like regrabs don't work randomly but there are a few factors to take into consideration when going for regrabs which can make the difference between getting the regrab and missing it.
Zoom out the camera and make sure morpheel is heading down, or horizontal. If she is going upwards it will fail, if she is going horizontal it is more likely but would be slow, if she is going down you will get the regrab and it should be quick
Make sure morpheel isn't about to turn a sharp corner when you force unequip Iron boots because she will get away from you and you won't get the regrab
Don't attempt a regrab if pillar parts are falling around you, however it is rare you will be in this situation
She shakes her head 3 times before she throws you off, if you didn't have a chance for the regrab just sink down to refill your air and try again
It is possible to beat morpheel without zora armour while only using two waterbombs for air-fill. Althought morpheel can give you very bad RNG making it impossible for a two bomb you should be able to get a three bomb considering it is quite rare.
Firstly you want to jump-attack or quickspin the eye for the quick cycle on phase 1 and do the cutscene float with your first waterbomb.
When the cutscene has just ended press A to drop the bomb and get your first air-fill (If morpheel makes you drop it you won't refill your air so make sure you drop it with A) and then use Iron boots to swim over to the left and slightly towards morpheel (you want to predict where she is going)
Equip Iron Boots and clawshot onto her back. Hit her four times and then force un-equip Iron Boots to get off without waiting. Then sink to the bottom. It is very rare she is so high at this point you won't make it to bottom, if you think this is going to be the case don't hit her all the times and sink sooner.
Do Swim With Water Bombs again but this time start heading towards where you predict morpheel is going (Don't drop the bomb immediately) and drop the bomb when you are cloes enough for a regrab or if the bomb is going to explode (Around 6-7 flash). It is during this stage that you decide if she is too high to reach or low enough for two bomb, if she is too high simply sink back down and refill again
If she stopped to try and suck you in, you should drop the bomb and mash A to swim higher to be able to clawshot her. Also bear in mind if she sucks you in you can just take the hit, and then mash A afterwards and get the clawshot afterwards. However keep in mind that getting eaten does a heart of damage.
Once on her back hit her four more times and then force un-equip Iron Boots and regrab her (you don't need to put the boots back on). You can now deliver the finishing blow, If you have about 1/4 of your air meter left you should survive the cutscene of her death animation