Why does an aircraft's nose tend to pitch downward at higher bank angles?
The answer is that the aircraft doesn't actually pitch down. When banking (or rolling) into a turn, the angle of attack (or the angle between the center of the wing and the flight path of the aircraft) stays the same. Assuming no correction to maintain altitude is made, the aircraft will continue on the same angle of attack, just banked over by some amount. Due to the bank, the lift produced by the wing is no longer vertical, instead, it is directly perpendicular (at least for our purposes) to the wings. Therefore, if the airplane is in a 30 degree bank, then the lift vector will also be 30 degrees from vertical. Doing some vector analysis, then you can determine that if the angle of attack hasn't changed the new vertical component of lift will be about 86% of what it was in level flight. Therefore, the plane would begin losing altitude. This altitude loss is what you see as the airplane pitching down.