Hi Ulrich, can you give us a link to the gyro you are using as well as to the other ones you are considering? It would be nice for us to see what you're doing so we can follow along, or try it ourselves.
You will want to make sure the ESC you are using for the turret rotation does not have "brake" enabled (most do). With brake it will not want to change direction immediately.
The other problem I would think is that the standard turret rotation motors (Heng Long/Taigen) are very slow and would not be able to move fast enough. A high speed servo, modified for continuous rotation, might work better.
From your description it sounds like in theory the 3-axis gyro is all you need, the problem is in the mechanical implementation (if I understand you correctly). There is actually no such thing as a "6-axis" gyro, there are only ever 3 axis of movement. What they call "6-axis" just means essentially that it is more sensitive, but the functionality is going to be basically the same. However as you point out maybe one of the advanced "modes" available on the "6-axis" would work better in practice.