BC is composed of the form factor and the sectional density. There are no other factors involved. If the form factor does not change, the BC will not change for a given projectile of a certain fixed size and weight. The form factor for bullets may change with different air densities if the bullets have marginal gyroscopic or dynamic stability, as the yaw may be different. There will normally be a change in group size with any change in stability, but it will depend on the bullet design and start stability, some may show small changes, others much larger changes. Pellets are somewhat different in that any change in yaw will usually give a wide variation in group size. This is because they are aerogyro stabilized with a very small value of aerodynamic overturning moment. Hence, aerodynamic drag should not vary greatly and neither will form factor.
What does change is air density, which will change the retardation of the projectiles, it is not a change in the BC causing the projectile to slow down at different rates at the different altitudes.