general relativity - How does exactly time dilate in a gravitational field, and how is it judged by different inertial observers?

When I tried to answer the SE question, Clock on a pendulum, I faced an ambiguity regarding time dilation in the gravitational field. To make the argument clearer, I designed the simpler thought experiment below:

Two similar clocks (1 & 2) are located side-by-side close to each other at point $A$ in a uniform gravitational field $g^\prime$ at rest with respect to the lab observer ($A$). The clocks are synchronous and are simultaneously set in motion at $u^\prime$ in opposite directions perpendicular to the field. After reaching two equidistant points from $A$, say, $E$ and $F$, the clocks are read by the lab observer. It is anticipated that both of the clocks, regardless of how much they have been dilated, show similar numbers as measured by the lab observer. See the attached Figure.

Now assume an observer $B$ approaches the lab observer at constant $v$ perpendicular to the field direction. In the absence of gravitation, it is easy to prove that both clocks, despite undergoing different time dilation due to different relative velocities as seen by $B$, show similar times as passing points $E$ and $F$. In this case, however, $B$ asserts that clock 1 first reaches point $E$, and then it's clock 2 that passes point $F$. In other words, the arrival of the clocks at the assigned points is no longer simultaneous, yet the clocks show similar times as passing the points.

However, when we consider the scenario in the presence of the gravitational field $g^\prime$, circumstances become odd from the perspective of $B$. Using acceleration transformation [Eqs. (1c)], $B$ measures two different accelerations $g_1$ and $g_2$, respectively, for clock 1 and clock 2 as follows:

$$g_1=\frac{g^\prime}{{\gamma_v}^2 \left( 1-\frac{uv}{c^2} \right)^2}$$

$$g_2=\frac{g^\prime}{{\gamma_v}^2 \left( 1+\frac{uv}{c^2} \right)^2}$$

So that we always have $g_1>g_2$. But, as we know, the greater the gravitational field, the slower the clocks run, and vice versa. Therefore, it is anticipated that, from $B$'s viewpoint, clock 1 runs slower due to greater G-field ($g_1$); and clock 2 runs faster due to smaller G-field ($g_2$). On the other hand, as mentioned earlier, $B$ sees that clock 1 (the slower one) reaches point $E$ sooner than clock 2 (the faster one) does for point $F$. Therefore, it seems that the clocks do no longer show similar times as passing over the points! Where is the problem?

Recall that if the obtained $g_1$ and $g_2$ were replaced by each other so that we had $g_1, it would become more plausible that the reading of the clocks would be similar as calculated by $B$, however, this is not the case!

Moreover, assume that observer $B$ is outside (away from) the gravitational field all time. However, if one assumes that $B$ moves at $v$ sliding onto the surface of the earth/planet inside the G-field, the final result won't be affected much. In this case, assume that the clocks and observer $A$ are all located on the surface of the earth/planet contrary to what is shown in the figure.

All in all, is it true to say that the more G-field, the slower the clocks run? Or, maybe there are other parameters that can indicate the reverse, at least from the viewpoint of some specific inertial observers.