I will start with recommending an article, and I will follow up with describing the information in that article relevant to you.
Olaf Wucknitz, 2004
Sagnac effect, twin paradox and space-time topology - Time and length in rotating systems and closed Minkowski space-times
Wucknitz argues that the circumference of a rotating disk can be thought of as a Minkowski spacetime (with just a single spatial dimension) that loops back onto itself.
Wucknitz argues that it is the loop topology that gives rise to the Sagnac effect, not the (centripetal) acceleration. In support of that Wucknitz offers the consideration: when you reverse the direction of rotation the direction of the Sagnac effect reverses with it. That is: reversing the direction of the rotation does not change the direction of the centripetal acceleration, but the direction of the Sagnac effect reverses anyway.
In section 7 of the article Wucknitz mentions an experimental setup created by Ruyong Wang, Yi Zheng, and Aiping Yao, the setup is referred as 'fiber optic conveyor'.
(2004 article: Generalized Sagnac effect)
The optic fiber is set up to run around two wheels. So: some sections of the optic fiber run in a straight line from one wheel to the other, and then there are the sections of fiber that go around the wheels.
Light can be introduced into the fiber, counterpropagating, and when the conveyor is moving with a velocity $v$ a phase difference is measured, the phase difference is proportional to the velocity $v$ and the length $L$ of the fiber optic conveyor.
Wang, Zheng and Yao write:
[...] we have discovered that any moving path contributes to the total phase difference between two counterpropagating light beams in the loop. In a previous experiment using a fiber optic conveyor (FOC), we showed our preliminary result that a segment of linearly moving glass fiber contributes [...] to the phase difference [...].
Olaf Wucknitz writes:
The more interesting general case, where the fiber moves along any arbitrarily curved, possibly self-crossing, path, can be interpreted very easily in terms of a closed Minkowski space-time whose spatial oordinate is measured along the fiber. It now becomes clear (and was actually tested in the experiment), that the effect depends only on the length L, the speed v and the closed topology. The specific shape of the fiber in 3-dimensional space is not relevant at all.
Length of the conduit versus enclosed area
The fiber optic conveyor used Fiber Optic Gyroscope technology, which measures a phase shift.
With a ring laser setup a frequency shift is inferred from a measured beat frequency.
I assume that the length versus area considerations apply equally to both cases.
With a ring laser setup the enclosed area is usually square or triangular, and then the enclosed area is of course proportional to the length of the path followed by the light.
I suppose that can make it seem as if the measured shift is proportional to the enclosed area.
However, I think Olaf Wucknitz presents a strong case that the actual determining factor is the length of the path that the light follows, not the enclosed area.
On whether relativistic analysis is necessary
One of the unique features of the Sagnac effect is that it cannot be used to distinguish between pre-relativistic and relativistic physics. In both contexts the expected outcome is the same.
In the series of articles by Kevin Brown titled 'Reflections on Relativity' there is a section about the Sagnac effect.
Kevin Brown writes:
the Sagnac effect is not strictly a relativistic phenomenon, because it's a "differential device", i.e., by running the light rays around the loop in opposite directions and measuring the time difference, it effectively cancels out the "transverse" effects that characterize relativistic phenomena.
(Incidentally, I am very interested in the laser physics of ring laser gyroscopes, I would very much like to learn more about that.)