You didn't mention ship mass so I would estimate it as 100 million tons(10^11 kg).
If I would take you question literally then first thing I would think of is offset of barycenter. Basically if you make two bodies of equal mass orbit each other then they rotate around center of their masses that is right between them. If one body is lighter then barycenter would move towards heavier body.
Earth weights 6*10^24 kg, you ship weights 60 millions million times less, so barycenter would move only 1/10 000 of mm from Earth center of mass. Earth gravitational anomalies would affect orbit of the ship much more than that.
But I suppose you mostly care how much atmosphere would affect orbit. Obviously, the bigger the ship, the less surface to mass ratio, so atmosphere affects it less and less. Usually satellites don't go below 300 km, but a ship this big can go much lower, especially for short time.
Calculations are relatively simple. Air resistance force would be 1 / 2 * Cx * p * V^2 * Sx, p - air density, V - ship speed(~= 8 km/s), Sx - front surface(Width: 806m, Height: 300m = about 2.4*10^5 m^2), Cx - resistance coefficient(= 2, because speed is so big that we can consider that all interactions are inelastic). So it simplifies to 2.4*10^5 * 6.4*10^7 * p ~= 10^13 * p
At 100 km density is about 5*10^-7 kg/m3 so resistance would be 5*10^6 N = 500 ton(2 kg per m^2). Ship would decelerate 0.05 mm/s^2, and lose only about 0.27 m/s per revolution. 1 meter per second on LEO corresponds to roughly 2 km of orbit height so it would lose 0.5 km of height per rotation. But deceleration would speed up quickly as air density grows exponentially, so it would go down after 10-20 revolutions. Or you can accelerate ship with engines accordingly - 0.05 mm/s^2 is nothing for battleship.
But as ship slows down it loses energy. According to energy conservation this energy should go somewhere - it turns to heat. That would be about 100 KWt per m^2, that's enough to burn small parts like antennas and make ship look very bad.
If you move it a little higher - 120 km, air density becomes 4 times less, 140 km - 60 times less. So at 140 km it would lose only 100 meters per revolution and external parts should be OK, though I think paint would suffer.
Below 100 km heat from air resistance becomes too much and orbit decays too quickly. But if you have appropriate(force shield?) protection and powerful engines you can go as low as you want.
At about 150 km heat is not a big problem and a ship that big can orbit for days even without turning engines on. At 200 km and above air is not a problem for such ship.