How does the third not require assumption? 
Because you eliminate all other possibilities...
Not if you start eliminating from the bottom of the list...
If you do that, you must conclude that their is an unknown, unobserved, and undetectable force that distinguishes between pendulums and other objects, causing pendulums to swing as if the earth is rotating.
That is correct. It distinguishes between a pendulum and other objects for the same reason the rotation of the earth supposedly distinguishes between a pendulum and other objects.
The rotation of the earth does not distinguish between a pendulum and other objects. The pendulum is suspended from a point which it is free to rotate around; if the point is moved, it is unaffected by rotation, because a point has no geometry. The pencil is on the ground and moves with the earth, which is not rapidly accelerating or decelerating.
Precisely my point. Since the pendulum is swinging freely it is subject to the Coriolis Force. The pencil on the ground is not.
Static friction. If you perform this experiment in a vacuum (to negate air resistance on an object with so little mass) with a pencil hanging from the string, you would experience the same effect.
Well, I'm glad we're in agreement then. It is the mechanics of the pendulum that leaves it subject to the Coriolis Force.
However, as it has been claimed, the Flat Earth is not rotating and so no Coriolis force can exist for this model of the Flat Earth. As a non rotating Flat Earth can not cause a Coriolis effect, then we can use this experiment to prove that the Earth is rotating.
Note: This does not prove if the Earth is flat or not (i'll get to that).
So, if we can show that a pencil is not effected by a Force because it is not swinging, we can rule out an ouside force pushing on the pendulum. It must come from something the pendulum is doing that the pencil is not. This eliminates any "Celestial Force" acting on the pendulum form the rotating of the Heavens (or whatever).
And the only explanation then, of the pendulum, is due to the Earth rotation (Flat or otherwise), we can conclude that the rotation of the pendulum occurs because the Earth is rotating.
It is direct experimental proof that differentiates between a rotating and non rotating Earth (flat or round).
Now, onto why the pendulum proves a Round Earth:
On a Flat Earth, regardless form the centre of rotation you are, the pendulum will rotate 360 degrees with each rotation of the disk. The distance form the pole makes no difference to the rotational speed of the pendulum.
However, on a Round Earth, because of the angle that you are to the horizontal plane at the poles, the speed of rotation of the pendulum will vary. The distance you are from the poles will determine the rate of rotation of the Pendulum
Here is a difference between what would occur on a Flat Earth and a Round Earth with a pendulum that is appears to rotate due to the Coriolis effect.
As observations show that the rotation does indeed vary depending on the distance form the poles, we can therefore rule out that we are on a flat Earth and as the variance matches with the expected variance of a Round Earth it is very likely that we are on a Round Earth as we can not be on a Flat earth as this experiment shows.
There are 2 distinct effects that we are looking at there:
1) Pendulums appear to rotate and experiments designed to detect outside forces can find any. Therefore the Pendulum is not being influenced by outside forces and it must be due to the rotation of the Earth (flat or otherwise) because unless something is being influenced by an outside force, inertia states that it won't begin to move (and as inertia is used to explain the Force we feel as gravity on FE, we have to apply it to the pendulum too, as it can be show to have weight and therefore has inertia under FET)
2) The distance from the poles effects the observed apparent rotation of the Pendulum. As this is incompatible with FET, we can therefore rule out FET.
So, from these two experiment: Foccault's Pendulum, and the Pencil (or you could use a plum-bob) balancing experiment, we can conclude that there is no outside force acting on the pendulum so the Earth must be rotating and that the Earth is not a disk but must be a Spheroid due to the variance in apparent rotational velocity of the pendulum.
In other words: These experiment prove RET and disprove FET.