Shooting a .308 at a mile sounds most worthy, I hope you guys have a great time doing it.
Option 1, if you have an existing 12-twist or 13-twist 30" barrel target rifle is to shoot a topnotch 155 target bullet (Lapua Scenar, or Berger 155.5 Fullbore) at the typical 3000 fps
Option 2 would be to to use a 1-10" twist 30" barrel with a long throat and shoot Berger 230 Hybrids at the equivalent-energy speed of 2466fps.
With the rifle zeroed at 100 yards, at 1760 yards you need 101.5MOA of elevation for the 155.5@3000 and you need 95MOA of elevation for the 230H@2466. So make sure you have a good hundred minutes up from your 100 zero and you'll be fine. Speed on-target at 1760 yards is 875fps for the 155.5s and 997fps for the 230Hs
Shooting with iron sights would be straightforward. The standard aiming mark size for 1000 yard shooting is a black circle 48" in diameter. Making an 84" circle (7 feet diameter) at 1760 yards would provide an equivalent sight picture.
Note that for both iron sights and scope, you do not need to aim at the target itself. If the terrain allows it, you could set up an aiming mark above the target. For example if the 1760 impact point is on the face of a hill and you could find or make an aiming point that was 100' higher up, you could aim at that and you'd only need to add in which case you'd only need to have about 30MOA (over your 100yd zero) on your sights.
If tigrr could arrange for a convenient aiming mark 60-80 MOA above the target then nobody would have to do anything special to their sights in order to come out and shoot this.