Since you don't have the Lantern when entering Temple of Droplets, you will first have to acquire the Magical Boomerang to perform Boss Door Clipping.
To obtain the Magical Boomerang, you have to do the four Tingle Brother fusions:
- Tingle in South Hyrule Field, GP required (fuse AND talk to him afterwards to fully activate the quest)
- Knuckle in Trilby Highlands, GE required
- Ankle in Lon Lon Ranch, G> required
- David Jr. in Lake Hylia, GP required
These fusions will open up the trees in North Hyrule Field, allowing you to open the chest and obtain the Magical Boomerang.
You will need 4 Kinstones in total to do all Tingle Brother fusions, 2 of which are gotten in Cave of Flames:
- The first one after the rollobite room
- The second one in the final room before getting the Big Key
The remaining 2 Kinstones are gotten through RNG Manipulation after obtaining the Bow:
Since the entrance to Temple of Droplets can normally not be reached without the Flippers, you have to activate the Ocarina Glitch off of the house on the top-left side of Lake Hylia to walk over the water instead.
To open the door, you can either:
use Yadra's setup
After obtaining the Boss Key, save+quit and activate the Ocarina Glitch. Go back to the beginning of the dungeon and use the Magical Boomerang to clip past the Boss Door. In order to correctly clip upwards after triggering the screen transition on the right, you will need the correct sub-pixels: