The naive approach would be using a meter stick in the usual way: this method does not take into account a possible problem, that of the simultaneity of the measurement;
Well, the usual way seems to keep the two "meter stick" ends (or markings) in touch with the two ends (or elements) of the object under consideration sufficiently long (i.e. typically for much longer than the ping duration between these ends), so that this particular simultaneity problem does not arise.
There is however a problem with this naive approach, namely:
how to determine whether the opposite ends (i.e. either the two ends of the object under consideration, or likewise the two ends of the "meter stick") had actually remained at rest to each other while the "meter stick had been alongside the object", such that some particular distance value (e.g. "one meter") could rightfully be attributed to either pair at all.
Of course, that particular problem is rarely considered in "everyday technical" applications of (what's perhaps calibrated and sold as) "meter sticks"; but it's nevertheless a profound conceptual problem to experimental physicist.
But let's now turn to the simultaneity problem, i.e. specificly:
considering two separate participants who are (known to be) at rest to each other, how to determine in mutual agreement which indication of one had been simultaneous to a given indication of the other.
the next logical thing to do is setting up some detector in a point $C$ such that information travelling from point $A$ and $B$ will be received at $C$ in the same moment.
Right. (First to note: observations by any one participant, such as $C$, "at the same moment" are also called "coincident observations" of that participant.)
That's the setup described by Einstein in his coordinate-free definition of "(how to determine) simultaneity" from 1917 (
"Relativity. The Special and the General Theory."; chap. 8: "On the Idea of Time in Physics". (There exist earlier and independent descriptions of such a setup, for instance by Poincaré; but its definitive importance had apparently first been expressed by Einstein.)
Is there a way of doing this?
The question is (and it is an excellent question, as far as I understand):
How exactly to identify a suitable participant $C$ (or in Einstein's description rather: participant $M$) uniquely?,
because, in general, there can be plenty of different separate participants whose observation of some indication of $A$ had been coincident with some indication of $B$.
The specification given by Einstein is of course
that $M$ should be identified as "middle between $A$ and $B$",
that the distance ratios determined between these three participants should satisfy: $\frac{AM}{AB} + \frac{MB}{AB} = 1$
(which relates to your question title) and
implicitly: that $A$, $B$, and $M$ should (pairwise) have been at rest to each other (which relates to the discussion at the beginning of this answer).
In relativity such measurements are obtained via flashes of light.
Right. (That is, for addressing the entire list of measurement tasks above.)
Besides giving the simultaneity defintion as specific example case ("If the observer perceives the two flashes of lightning at the same time, then [...]") Einstein also noted that
"All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points." .
How do we know that this is the best method?
Because this method is the most perfectly unambiguous and comprehensible we know.
If anyone reports "I have seen a particular flash indication of $A$, and a particular flash indication of $A$; and my very first observations of these two flash indications had been at the same moment." then we can comprehend outright that report; and we may grant that anyone reporting that has at least the comprehension "to know what he's talking about", just as you and me.
(Contrast this with other, perhaps "more practical" or "more traditional" methods where we and anyone is supposed to understand outright which clocks were and remained "good" and/or "undisturbed", which clocks had been "transported slowly", which artefacts had "preserved their shape", who "felt free", or somesuch.)
Now at least to begin to address the above list of measurement tasks by means of the relativistic method based on signalling and judging coincidence (or else: sequence) of one's observations, we can require that
for any indication which $M$ had stated $M$ observed $A$'s corresponding reflection indication coincident to observing $B$'s corresponding reflection indication,
for any indication which $A$ had stated $A$ observed $B$'s corresponding reflection indication coincident to observing that $M$ had observed that $A$ had observed $M$'s corresponding reflection indication,
for any indication which $B$ had stated $B$ observed $A$'s corresponding reflection indication coincident to observing that $M$ had observed that $B$ had observed $M$'s corresponding reflection indication.
(These conditions are still not sufficient for establishing that $M$ had been at rest with respect to $A$ and $B$, and thus for identifying $M$ uniquely as the "middle between $A$ and $B$". But at least this illustrates all conditions which can properly evaluated by $A$, $B$, and $M$ themselves. Further conditions necessarily involve additional participants.)
Finally, the length attributed to the pair $A, B$, as characterizing their geometric relation to each other (besides having been at rest to each other) is expressed in terms of the ping durations between them (i.e. $A$'s duration for any indication until observing $B$'s corresponding reflection indication, or likewise $B$'s duration for any indication until observing $A$'s corresponding reflection indication):
$$AB := \frac{c}{2} \tau A_{\circ}^{\circledR B} = \frac{c}{2} \tau B_{\circ}^{\circledR A},$$
which is therefore indeed a distance, rather than a quasi distance;
and where "$\frac{c}{2}$" is foremost a symbolic, non-zero factor for distinguishing ping durations from other incidental durations which may be determined. Due to the definition of "(how to measure) speed", the symbol $c$ is only subsequently identified as universally representing "signal front speed".
