Lana using Hasty Lana Hop is faster than Master Sword Link here. Any movement around the map should be Lana Hops to save time.
Run to the Settlement Entrance Gate to the west to make it close.
Head back toward S. Desert Keep, defeating each of the captains along the way.
Defeat the Lizalfos inside and take the keep.
Repeat this for South Oasis and Central Keep.
The Lizalfos in the 2nd and 3rd keeps may not be in them when you get there. Defeating them will make it less likely that they will give you keep peril text later, but will waste time now.
After you have taken all three desert keeps, head to the Settlement Entrance that closed on you at the beginning.
Now take the N. Settlement and S. Settlement keeps.
Hop across the map to the hallway south of Sacred Grounds.
Once the door opens, kill each of the four Aeralfos. One or two will probably run away down the west hallway, this is fine since you will head that way after.
Run through the east hallway, defeating any Aeralfos that tried to run. Head toward the area east of South Oasis where Dodongo spawns.
Grab the green magic pot in Arbiter's Keep as you pass through.
Once you reach Dodongo, if he doesn't open his mouth immediately, use the magic to expose his weakpoint.
Defeat him using the triple walls C4 combo.
Head to Manhandla next, followed by Gohma, defeating them using the same combo.
If Manhandla doesn't open up immediately and you still have the magic, use it on them.
Ideally there will be minimal keep-in-peril text and ally-in-trouble text playing through this part of the level.