Dave,
Everything rotating body will display "gyroscopic precession". Doesn't matter how fast/slow it turns, doesn't matter it's mass, mass distribution, etc.
It gets too complicated when multiple bodies get together to form a rotating system, so it's best to ignore GP. In fact also ignore aerodynamic precession...it just confuses the issue by generating argument about which precession theory applies to helicopters. Neither one does. They are both convenient explanations.
Examine the theory behind GP or AP. You will find it is the same rotational dynamics behind each. The blade wants to flap at it's natural frequency. If the natural frequency is the same as the rpm of the rotor, then the apparent lag is 90 degrees. If the natural frequency is higher, the lag is lower.
A gyroscope can be considered to be a bunch of point masses that are spinning around an axle and are free to "flap". Each point mass is just like a high school physics pendulum. The natural frequency depends on the length and the restoring force, sq. rt. of g/l in high school physics. Since the restoring force for the point mass is the centrifugal force (it's okay to use centrifugal force...just not a real force) then the force depends on the distance from the center, and the speed of rotation. In the end, we find each point mass has a natural frequency that is exactly the same as the rotational speed of the gyroscope. Reading the above paragraph, that means the apparent lag on a gyroscope is exactly 90 degrees. This shows that GP is just a special case of the blade frequency theory that is a much better way of looking at helicopters. QED
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The full beer can with a slowly spinning gyro will do what you want, but it will require a small corrective force to keep the intended tilt in the right direction. That small corrective force corrects the gyroscopes reaction to the impressed moment.
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Lu, with the propellor example, the engine moves in response to the planes movement, that energy is absorbed by mounts, but some still transmitted to the aircraft. IIRC there are some aircraft with reversing gears between the engine and propellor to get closer to a zero angular momentum condition.