Welcome to physics.SE! Cool question.
Yes, this works. In fact, there was a classic test of special relativity done using exactly this technique, Bailey at al., Nucl. Phys. B150(1979) 1. They accelerated muons in a circular accelerator to $\gamma\approx30$, and found their half-lives to be lengthened by exactly that factor. For a more detailed description of the experiment, see example 6 in section 23.2 of this textbook, which I'm the author of.
One of my problems with this concept is that a rotating radius would have different speeds along its length, possibly therefore leading to different degrees of time dilation along its length.
I see. I guess you were imagining spinning a macroscopic object about its center of mass...? That's (a) impractical to do, and (b) much more difficult to interpret, as you've correctly figured out. The particle physics experiment evades this issue by using subatomic particles, which have a definite value of $r$.
I quickly decided that moving forwards and backwards wouldn't work—the speed would be zero at the turning points.
There have actually been tabletop experiments done in this style, with atomic clocks. The effect can still be calculated even though $\gamma$ is varying. The velocities are $\ll c$, but the atomic clocks are extremely precise, so you get a measurable effect. For a brief description, see example 5 immediately above the one I previously mentioned. The reference is Chou et al., Science 329 (2010) 1630.