Consider a gyroscope which is hanging with a string. Is it possible to $flip$ the orientation of the a gyroscope by oscillating the point of suspension? How does it come out mathematically?
Short answer: flip not possible.
Take a gimbal mounted gyroscope, spinning fast. And now you want to reorient it, so that in the end position the angular momentum vector is reversed (orientation flipped). That is possible with precisely the right torque, and you stop supplying that torque when the desired end position is reached.
You get the kind of control you need only with a torque.
(To remove any possible ambiguity: Here I mean by 'torque' a narrowed-down case: a moment of force such that the torque vector goes through the center of mass of the gyroscope wheel. Also called a 'couple'.)
I don't see any applied oscillation having the effect of flipping the orientation. On the contrary: an oscillation goes back and forth, I expect its overall effects to cancel out. For instance, if you apply an oscillation, with gradual buildup of the amplitude, and you also go out gradually, then I expect that the before and after orientation will be the same.
So: for flipping the orientation an oscillation is pretty much the method most certain to be unsuccesful.
This raises the question, why did you even consider the idea that some oscillation might possibly flip the orientation of a gyroscope?
Perhaps, I don't know, your thinking was that given that gyroscopic precession seems very counter-intuitive, maybe there is also a very counter-intuitive response to an applied oscillation?
Further reading: discussion by me of gyroscopic precession in a physics.stackexchange question from 2012