Will the plane of oscillation of a pendulum change if I rotate the suspension point

Actually, the idea for the setup we call 'Foucault pendulum' occurred to Leon Foucault in the following manner: he had clamped a long rod in the chuck of a lathe, he twanged it, and when he rotated the chuck the plane of vibration did not rotate. Foucault surmised that the momentum of the vibrating motion is not in any way bound to the physical orientation of the rod. In effect the momentum of the vibrating rod was behaving in accordance with Newton's first law. Unless acted upon by a force the momentum vector of a moving object keeps pointing in the same direction.

I did a search on youtube, and much to my delight I discovered a youtube video for this subject, titled: 'Scientic contributions of Leon Foucault', by dr Matt Nehring. Dr Nehring clamps the vibrating rod in the chuck of a drill

I assume that the reason that Foucault ended up with a gravity pendulum was that a gravity pendulum allows a setup with a large amplitude and large period of oscillation.

Any oscillation will have the property that the orientation of the momentum of the oscillation is not in any way bound to the physical matter that is vibrating.

The rotation state sensor in a smartphone has this type of rotation state sensing as its operating principle. It's not necessarily a vibrating rod, but some elastic part is oscillating, and the momentum of the oscillation tends to keep pointing in the same direction. (Depending on the build quality there will be some drift, but for the purpose of assisting navigation the sensors are quite sufficient.) The rotation state sensor in a smartphone is generally referred to as 'the gyroscope' but it isn't a gyroscope in the sense of consisting of some spinning wheel, suspended in gimbal mounting. The rotation state sensor of a smartphone is of the vibration type.

About why a larger Foucault setup is better:

The Foucault effect is very small, it's easily swamped by other effects. For instance, if the wire has a direction of bending where it is less bendable - and hence will have a direction of bending perpendicular to that where the wire is more bendable - then the force that tends to push/pull the pendulum bob back to midpoint will not be exactly the same in all directions, which would mean that the period of oscillation would not be the same in all directions, and that has the potential to swamp the Foucault effect. All the effects that unless prevented will swamp the intended effect are collectively referred to as 'parasitic effects'. In the case of a Foucault pendulum: the larger the pendulum, the less troublesome the parasitic effects.