If you lose all your shared lives and die, you will respawn at the top of the tower outside. You cannot back out of this story quest. If you want to continue the game, you have to successfully kill Lothar.
Go into this fight with three lives, full armor and health, and as much of a full load of ammo as possible.
The first phase of the boss fight has Lothar flying around with a jetpack. The only thing that will damage him is to target the jetpack. You need to shoot when his back is to you. Go up one of the elevators and circle around to try and target Lothar's backpack. His armor is the square type, so the assault rifle and pistol work on damaging him. Save your heavy weapon ammo for the second phase. Keep jumping over to the mounted heavy weapons. The laser weapon will damage his armor. He will drop down to the arena level periodically. Leave the mounted weapon and shoot him with your assault rifle.
When Lothar's health and armor bar is depleted, he will fly high in the air and then crash at the arena level. Before going to his crash site, load up on ammo, armor, grenades, and health. Kill all enemies before investigating the crash site. The enemies will not keep spawning. At the base of one of the four pillars, there is a door that can be opened with the electrokraftwerk weapon. Inside that room is ammo, armor, health, silver coins, and grenades.
Don't do what I did. I had not figured out the boss fight mechanics at first. I did not clear out all the enemies before going to his crash site, and I got wasted by Lothar and the other enemies. I lost all the shared lives and died. When you respawn after dying, you are stuck with the ammo you had when you died, and one shared life. To help you get ammo, when you respawn, run around the area and loot all the ammo. Open up the electrokraftwerk door and stock up. Then jump off the tower twice and die. You may need to rinse and repeat this process a few times to get enough ammo. When you are ready to kill Lothar, respawn, and you will have a full load of light weapon ammo and a some heavy weapon ammo. That's the best you can get.
Go to the crash site and immediately go the center of the arena and move around the center, which will give you cover. During this phase, Lothar has air support and the aircraft strafes the arena with missiles. If you get low on armor or health, go to the pillars and go up to find some health and armor. Also, the room that Lothar comes out of (the crash site) has armor, health, and ammo.
In the second phase of the boss fight, Lothar's armor is the dash type. When he comes out of the crash site, the ideal situation is to unload ten shots from the uberhammer. Aim at the floor right in front of him and the splash damage will nail him. If you get ten uberhammer shots on Lothar, most of his health and armor will be gone. Use a laserhammer, shotgun, or submachine gun to get the rest and kill him.
The faster you kill Lothar, the better. The aircraft missiles will down you very fast. And if you are playing solo with the AI like I did, good luck. Your stupid sister loves to get blasted in laser beams and bullet streams, and not get into cover.
Update on September 6, 2019. Bethesda had an update today that supposedly fixes some of the problems with respawning after dying and having only the ammo you had when you died. So this boss fight should be better if you have died and had no ammo.
From their update notes:
"Players who expend all lives and reset to the latest checkpoint will now find their base weapons refilled to at least 50% of ammo capacity. This will function with all ammo modifier skills and upgrades but does not affect special or heavy weapons."