Do clocks measure conformal time (new argument)? Assuming the spatially flat FRW metric for simplicity:
$$ds^2=c^2dt^2-a(t)^2(dx^2+dy^2+dz^2)$$
where $t$ is cosmological time, $a(t)$ is the scaling factor and $x,y,z$ are co-moving spatial Cartesian co-ordinates.
Light freely propagating along the $x$-direction follows the null geodesic with spacetime interval $ds=0$. Therefore for small finite co-ordinate intervals $\Delta t$ and $\Delta x$ along the null geodesic we have the relationship:
$$\frac{c\ \Delta t}{a(t)}=\Delta x\tag{1}$$
The co-moving interval $\Delta x$ in this expression must be constant as it is the distance between co-moving spatial co-ordinates. This implies that the cosmological time interval $\Delta t$ must scale like $a(t)$.
Now let us imagine a model of a clock that consists of a rigid ruler of fixed proper length $\Delta l$ with an optical fiber attached along it.
Let us assume that the rigid ruler is co-moving with the expansion of the Universe.
Let us suppose that each tick of the clock consists of a light pulse sent down the optic fiber (assuming refractive index 1 for simplicity) which takes a constant time interval $\Delta\tau$ given by:
$$\Delta \tau = \frac{\Delta l}{c}\tag{2}$$
Now the co-moving interval $\Delta x$ in the null geodesic of freely propagating light, equation $(1)$, and the proper length of the rigid ruler $\Delta l$ are both constant. Without loss of generality let us arrange for them both to be equal in magnitude so that we have:
$$\Delta x=\Delta l\tag{3}$$
By substituting equations $(2)$ and $(3)$ into the null geodesic equation $(1)$ we obtain the relationship:
$$\frac{\Delta t}{a(t)}=\Delta \tau\tag{4}$$
Thus each tick of our clock $\Delta \tau$ does not measure an interval of cosmological time $\Delta t$ as one might expect but instead it measures an interval $\Delta t/a(t)$ which is in fact an interval of conformal time.
Is this reasoning correct?
 A: Firstly, notice that the Frequency of light used to measure time will not remain constant. I wont use this but do take note. 
Secondly notice that If the object is rigid, then and One end is fixed at x = 0, then the other end in the co-ordinate systems used will be at
$\Delta x = \frac{\Delta L}{a(t)}$
So Now light using equation 1 from your question we get 
$\Delta t = a(t)*\frac{\Delta x}{c} = \frac{\Delta L}{c}$
So both clocks reproduce cosmological time as expected. 
I hope I understood your arguments correctly. I think the flaw in the argument is you cannot allow for both the rod to be rigid and its ends to be co-moving at the same time.
A: Do clocks measure conformal time?
No. I would venture to say that clocks don't literally measure time at all. A clock isn't some cosmic gas meter with time flowing through it. What clocks do is "clock up" some kind of regular cyclical local motion and show you a cumulative result that you call the time. A pendulum clock clocks up the swings of a pendulum. A quartz wristwatch clocks up the vibrations of a crystal. Your light clock clocks up how many times light moves back and forth, just like the parallel-mirror light clock. And in all cases, the clock rate is affected by macroscopic motion. Keep parallel-mirror light-clock A on Earth and send B on a fast out-and back trip, and the clock B reading is less than the clock A reading. Because the local motion is reduced by the macrosopic motion, because the maximum motion is c:

Public domain image by Mdd4696
Imagine you're panning your gedanken telescope and watching that local motion in clock B: it's slower. The spacetime interval is the same for both clocks because the light-path lengths are the same. The ds=0 only tells you that the motion of light less the motion of light is zero. We talk about time dilation, but there is no actual thing call time being stretched, just rates of motion being reduced.
That's what happens with gravitational time dilation. See Einstein talking about the speed of light being spatially variable here. Nowadays we say the coordinate speed of light varies with gravitational potential, or spatial energy density. And when it comes to conformal time we're talking about 46.9 billion years, wherein "the particle horizon is equal to the conformal time η that has passed since the Big Bang, times the speed of light c". Only we liken the early universe to a black hole, and swap space and time such that the expanding universe is like pulling away from a black hole. Only at the event horizon the coordinate speed of light is zero. So we can reason that the speed of light isn't constant over space and time, so multiplying by c doesn't work, and at the time of the big bang your light-clock wasn't ticking. As for it being co-moving, let's simplify matters by making it stationary with respect to the universe. As the universe expands the clock starts ticking, slowly at first, then faster. It's motionless with respect to the CMB rest frame, and it now shows a clock reading of 13.8 billion years. Not 46.9 billion years. So it isn't showing conformal time. 
What's it showing? The universe is flat now and always has been, which is why you use the flat FRW metric. See what Mortimer said on Physics Forums: proper time is the time as measured by a clock in motion or in a gravity field, while coordinate time is the time as measured by a stationary observer at infinity where there is no gravity. And there is no overall gravity in the universe. So your clock is showing coordinate time. Even though for all you know, it's been around for ever. 
