If you're Item Level 115+ and have Rhapsody in Mauve, it may behoove to do the Tavnazia Voidwatch missions now. You can use Trust and buy Phase Displacers for 1000 gil each (or, if you're very strong, kill them without). It will cost 20k to burn through the four prerequisites in a little under half an hour, which will open up the option to warp directly to Purgonorgo Isle.
While you're not likely to be able to get these, the following event-related RSE swimsuits have an enchantment that can teleport you to the isle if you've visited Bibiki Bay previously. You do not need to have visited the isle. Some holiday events let you purchase them from an Event Moogle.
Do note that retrieving an item with a reuse-timer from an Event Item Storer will result in a reset of the timer to its maximum amount. In other words, these can not be used before 20 hours have passed from retrieving it.
If you don't want to or are unable to do the above, you'll have to take the Manaclipper.
First take either the Unity Concord warp to Bibiki Bay (level 119), the Survival Guide if you have it, or walk there via Buburimu Peninsula, then buy a multi-ticket from Tswe Panipahr (you'll have to come back again and nothing's as bad as being bounced by accident). Talk with Fheli Lapatzuo and if she says they're going to Purgonorgo Isle, autorun to the end of the dock and AFK for about 15 minutes plus her estimated arrival time. You could warp and do other things, then take the Survival Guide back in under two minutes, but it's not worth the risk. Otherwise, set an alarm for 5 minutes plus the time the next ferry arrives, go AFK or do other stuff, then after it leaves, auto-run and AFK. Make sure you check which destination the Manaclipper is going to before auto-running.
Manaclipper schedule on that dock is Purgonorgo Isle > Maliyakaleya Reef Tour (10m) > Purgonorgo Isle > Dhalmel Rock Tour > repeat. The relevant boarding and departure times are 4:50-5:30 and 16:50-17:30. Consult Manaclipper/Schedule for the full schedule.
The fastest method is the Survival Guide in the zone. If you don't have it, the other options are either to walk there from North Gustaberg or to use the Survival Guide to Gusgen Mines (or Teleport-Dem and head east), then go west and follow the right wall north and east. Since you'll probably end up using the ENM warp, North Gustaberg may be a little faster than running from the north end of the zone.
There is also a proto-waypoint that lands you right near Twinkbrix .
If you are planning to use this method, (which seems like the easiest and cheapest) buy or farm 2 per person, since you will be returning to the good old shaft again... (you can optionally farm/purchase 1 additional Snow Lily, as you will need another for Mission 7-3 Fire in the Eyes of Men)
Note: Recent updates have added a Home Point outside of the entrance to Mine Shaft #2716 allowing you to only require one Lily, assuming you attune to the crystal on arrival. You'll need to exit the area to access the Home Point prior to the fight as you get warped out and away from the Home Point after battle.
Give Twinkbrix an amount between 1-10,000 gil, to have a chance at getting the Key Item Shaft Gate Operating Dial, which is required for the two ENM60 battles here. He will roll a number 2 through 100. Your chances of winning is based on how much money you give him. 10,000 gil will only give you a 50% chance of winning.
Once you have the Key Item, you need to pay another 2000 gil to be warped to the Mine Shaft #2716.
Note: If you want to do one of the ENM60 battles while you are here, Sleakachiq, just outside the exit, will warp you back to the Mine Shaft 2716 for a nominal fee, of course.
All Moblins possess two-hour abilities for their job and all have the ability to call Bugbby to the center of the Battlefield, ignoring any kiting in progress and resetting hate. Whoever is kiting Bugbby should make sure to reclaim hate when this happens.
Each player doing this mission will need a Gold Key. They can be obtained by defeating certain Moblins and/or Golbins in Newton Movalpolos. They can also be sold in Bazaars and traded, but cannot be auctioned in the Auction House.
If you disconnect while in this cutscene you will lose key but can recheck the door to repeat the cutscene.
If you are in a party (CoP static, for example) and helping with the battle from the previous phase of the mission, do not trigger this cutscene before you enter the battlefield. The cutscene spits you out at the battlefieldexit in Newton Movalpolos, and you will not be able to Escape and teleport back to Mine Shaft #2716 if you have just used Twinkbrix to get there. But you can pay 800 gil to the Moblin Sleakachiq, just outside the exit and choose to warp back to the mine shaft entrance.