The key sentence is "from the frame of reference of the accelerator, the circumference must stay the same." So from the accelerator frame, no length is changing, just accelerator's time passes by more quickly by a factor of gamma than in the particle's frame. This is what allows the particles to "travel extra distance" than would normally be possible in Newtonian mechanics. So that is the explanation from the accelerator frame.
From the particle's frame of reference, time passes by normally, but the distance between each particle will be Lorentz contracted. Imagine a particle directly in front of it. Since they're traveling in "almost" the same direction, there won't be a whole lot of length contraction since they're pretty close together. But now imagine a particle at 0 degrees and one at 90 degrees (a quarter way around). One is traveling in the "x" direction and the other in the "y" direction. Since the first particle traveling in the x-direction has no velocity in the y-direction, there will be significant length contraction, and actually the exact amount contracted would be the Lorentz factor $\gamma$. Now two particles spaced on opposite sides are traveling towards each other, so length contraction is further exacerbated. Since all particles are evenly spaced, on average there is a factor of $\gamma$ of length contraction, so the circumference in the particle's frame appears to be $C/\gamma$.
So all in all, in the accelerator frame, the circumference is still normal, and the physics checks out since the particles are in a different time frame, allowing them to travel all the way around. In the particle's frame of reference, the circumference appears shorter, so they are able to make it all the way around. Since the radius is perpendicular to the motion of the particles this will not be Lorentz contracted. This is known as the Ehrenfest paradox.