Temple of Droplets is one of the Dungeon that gets completely destroyed by the glitches you can use in here. Thanks to Boss Door Clipping and the Ocarina Glitch, it is possible to completely skip the left and right side of the dungeon after getting the Boss Key. The Lantern is still required, however, as there is no method to defeat the boss without it.
After getting the first small key, you can use the Ocarina Glitch to get the Boss Key without using a small key, saving it for the mini boss.
Using Boss Door Clipping and the Ocarina Glitch, it is possible to reach the Blue Warp Room early, which allows us to completely skip the Flippers in this dungeon. More info can be found here.
Using the Ocarina Glitch, it is possible to skip everything after the Lantern and go straight to the second lever to activate the boss.
There is a manipulation possible for Big Octo to always get 0 ink on the third phase.
Without manipulations, here is how the fight can go:
1st phase: Octo will always shoot boulders at you until he's hit 3 times.
2nd phase: the pattern is suck, boulder, suck, boulder, boulder and repeats that pattern until he's hit 3 times.
On his 3rd phase, it's completely RNG based:
Before it performs each attack pattern it will have a 25% change to charge attack the wall making rocks fall, this can chain multiple charges together.
Octo has 3 attack patterns to choose from:
A) Rock>Move>Rock>Ink
B) Suck>Move>Rock>Ink
C) Rock>Move>Rock>Move>Rock
When deciding what to do, octo will select a random pattern from the following list: [A, B1, C, B2].
(B1 and B2 are both pattern B but numbered for clarity). Octo will never select the same entry in this list twice in a row (at the start of this phase it pretends the last pattern was A), if it randomly selects the same pattern as the previous cycle then it will select the next entry in the sequence instead (If B2 is selected it wraps back to A).
The following Chart shows the probabilities of each pattern at each stage of the fight.
Since we are guaranteed to get at least a single rock each cycle, 2 ink is the worst luck if you reflect all boulders he shoots at you. However 1 ink is by far the most common total.