Just finished this in a couple hours and wanted to share my method. I found this a lot easier than Hard, dying only 6 times in this mode versus 106 times in Hard mode, and 5 of those times were me accidentally dropping a Supply Drop on myself.
For the whole run I used maxed out Energy Shield, Reflex Booster, Dash, Use of Weapons, and Resource Optimizer. By the end I had added Supply Drop, Power Attack, and Grid Converter. Could probably swap out Reflex Booster for Power Attack and Supply Drop at the beginning.
My strategy was basic: Energy Shield + Dash with a strong melee weapon from Supply Drop (personal favorite is Vis Carver). That got me through the whole game except the Twin Sisters, where I traded Energy Shield for Overload. Dashing into an enemy with the shield up, if it doesn't kill them outright, which it will with a lot, will stun them and give you plenty of time to finish them with a melee hit. For all the arena fights I basically dashed around the level with the shield up sending enemies flying everywhere. This also works on bosses; Shield, Dash, stun, melee charge attack, back off, rinse and repeat. You can also time Supply Drops to fall on bosses while stunned, dealing a good chunk of damage; this was especially useful in the TrafficKing first fight, which gave me the most trouble in Hard.
And that's it. 3 or so hours, 6 deaths (5 self-inflicted), and 100 GS in the bag.