What's the deal with MOND?

Problem: Assume (don't question) you know for certain that

$$F=ma\mu(\frac{a}{a_0}),$$

where $\mu$ is an unknown function. (I know how you feel, but put those feelings aside and think about it as a solely math problem😅)

Your goal is to determine the unknown function $\mu$. 

Another assumption is that $a_0$ is found by the following:

In an expanding Universe

$$ds^2=-c^2 dt^2 + R^2(t) dx^2,$$

(note that as a special case, for $R(t)=1$ this yields the Minkowski metric)

Then $a_0$ is just the acceleration of null rays of the above metric, meaning that if you let $ds=0$, then

$$cdt=\pm R(t) dx$$ so $$\frac{dx}{dt}=\pm \frac{c}{R},$$ then $$a_0 := \frac{d^2x}{dt^2}\Big|_{t=0}.$$

The simplest approach to solving this is to recall that, if we define Force as

$$F=m\frac{d^2 x}{d\tau^2},$$ where $d\tau=\frac{ds}{c}$, by chain rule we have $$\frac{d^2 x}{d\tau^2}=\frac{d^2 x}{dt^2}(\frac{dt}{d\tau})^2 + \frac{dx}{dt}\frac{d^2 t}{d\tau^2},$$

So by comparison with the assumption

$$\mu=(\frac{dt}{d\tau})^2,$$

but this shows the problem: If we use the metric above, $\frac{dt}{d\tau}$ is not a function of acceleration.