I don't think you can assume that, because a Born rigid congruence in the presence of a gravitational wave cannot possibly be a geodesic congruence; if it were, then LIGO would give a null result and would not be able to detect GWs.

Most of the curves in the congruence won't be geodesics. But the reference congruence can be anything that makes sense, it represents the worldline of the "observer".

Probably not all the way to B=level, but - let's take an example of the globe. The 3-d analogy to Fermi normal coordinates would be to pick a particular point on the globe, and measure your bearing (which picks the direction you start out in) and the distance you move in that direction. Then you basically use the distance and bearing as polar coordinates on a sheet of paper (which, unlike the globe, is not curved.) The flat paper has whatever additional coordinates you like on it (say, a square grid of cartesian coordinates). The purpose of the construction is to make a map from the curved manifold of the surface of the Earth to a flat manifold (the piece of paper). The bearing/angle is an intermediate step in the contruction, not necessarily the end result. Another non-B level remark, mathematicians call this the exponential map, https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry), it's a mapping from the tangent space at a point to the manifold itself. The tangent space is the "flat sheet of paper", the manifold is the curved surface of the Earth.

In 2-space + 1 time, rather than a reference point, you have a reference worldline. At any instant in time (event on the worldline), you can construct an infinite number of spatial geodesics that radiate away from that point. You choose the subset of those that start out orthogonal to the reference worldline, that is a 2d space. Then you use the same principle of distance/bearing to map points on the space-time manifold to points on a flat plane. The time coordinate for any point on any of the curves reached via the radiating geodesics is the proper time reading of the clock on the observer on the reference worldline. The space coordinate are given again by the distance and bearing.

Time dilation is given by the usual procedure of the ratio of proper time to coordinate time. I'm not sure how to B-level that, but that's how time dilation in , say, Schwarzschild coordinates works.

There's several non-B level papers you can find with google, and MTW has a discussion of these coordinates.

What if instead of putting A and B close to each other we set them up on the opposite sides of Earth? Would not we be able to detect a wave passing through by noticing their signals temporarily going out of sync due to different distance to the source?

Depends what you're trying to do. If you're just putting mirrors a really long way apart and bouncing signals between them then yes that would be more sensitive. It's just an upscaled LIGO, with a few minor engineering challenges to do with tunnelling through the core. You might be better to put the mirrors out in space, which is a future design I believe.

If you are planning to use the Earth as a rigid bar sensor, however, then moving the clocks further apart makes the problem worse, not better. The speed of sound is so low in rock that one side of the Earth doesn't "know" the other side is in motion until hours after the gravitational wave has passed. So your clocks would just acts like free-floating clocks and you don't get any accumulated time difference from it. Just the transients that a LIGO-style detector sees.

Transients is what I'd be looking for.
Seems like the idea is theoretically viable. Feasibility of practical application requires extensive research though.

By transients, I mean you could in principle, replace LIGO mirrors with clocks. Due to the distance change associated with the gravitational wave each clock would see the other's rate wobble. LIGO does this much more precisely with interferometry, and the practical difficulties of using clocks instead are, as V50 pointed out, way, way beyond anything you are imagining. Whatever it is you are imagining.

OK, here is my best shot at a truly B-level explanation of my remarks, sorry for the delay.

General Relativity can be understood as drawing a "map" of space-time. It's a 4 dimensional map. Sometimes this is called "the block universe". GR has a mathematical entity called the metric. This metric can be understood as representing whether or not the map is to scale - it also tells about other sorts of distortion of the map.

If you draw your map to scale, it's easier to interpret it.

Fermi-Normal coordinates are a technique for drawing maps that are to scale and not distorted in a small region around some specific observer (who is represented on the map by a worldine, the worldine is the timeline of the observer)., Because it is drawn to scale and undistorted in a small region, the "map" is easy to interpret. . There's more to it than that (and the language is rather vague) but that's the executive B-level summary and motivation for using Fermi-Normal coordinates.

Note that it's impossible to draw a map of a curved 2-d surface on a flat sheet of paper that is everywhere to scale (for instance, making a map of the surface of the Earth that is undistorted). Similar remarks apply to drawing a map of a curved 4-d space-time on a "flat" 4-d manifold.