If I tie a rope to a clock and spin it over my head, would it lose synchronization with a clock sitting at my feet?
Yes, the clock you're spinning will lose synchronization. If you spin a clock around on a rope very quickly, it will run slowly from your point of view and be behind your wristwatch when you compare them. This is essentially the twin paradox. However, if you throw the clock up in the air and let it freefall, during freefall it will actually run faster than a clock you're holding. This is because the clock in free fall is following a shortest path in curved spacetime, and the "Principle of Maximal Aging" (a neologism of Taylor and Wheeler in their relativity book, I believe) applies.
These effects, though, are not due to the forces on the clock. Time dilation in special relativity has nothing to do with the forces on an object. It is a purely kinematic property.
Observing from your own inertial frame, if you the know speed of a clock relative to you, that doesn't let you deduce the force on it. There could be many possible forces resulting in the same speed. (For example, if the clock moves in a circle its speed is constant but there's a force on it. If it moves in a straight line its speed is constant and there is no force.) Nonetheless, the time dilation from your frame is completely determined by the clock's speed. Thus, time dilation doesn't have to do with whether there is a centripetal force. It only has to do with the motion itself. It's calculated from first derivatives of the coordinates, while forces relate to second derivatives.
The basic relativity you learn about in an introductory course (light clocks and spaceships with meter sticks, etc.) doesn't tell you how things behave - what the rules for fields and particles and things are - it only tells you the properties of space and time. That's what I mean when I refer to "kinematics". You can then make theories of physics, theories that tell you what fields and particles do, that respect the laws about spacetime. Time dilation, though, is derived completely from the kinematic part.