Clock postulate validity As far as I understood, the generalized clock postulate says that there is no difference between the measurements of a non-inertial observer  (NIO) and those of his co-moving inertial observer (CMIO) who instantsneously shares his time and position (of course the CMIO will change after an infinitesimal time interval).
However: let's suppose that in this time interval both the NIO and the CMIO measure the acceleration of a particle that is passing near them. Basing on classical mechanics, I would expect the two measurments to be different, since NIO's also depends on the fictitious force caused by its own acceleration, while the CMIO's does not. Where am i going wrong?
 A: The postulate does not tell you how your clocks will behave in accelerated frame. It  tells you how well made clocks should behave.
The fact that acceleration has influence on physical laws means, that mechanism with which the clocks are working will most probably be also influenced by the acceleration and this will influence the readings. However, the manufacturer of clocks must make sure, that this influence is negligible within the stated accuracy in the range of accelerations for which the clocks are built. I.e. it is on the manufacturer of the clocks to make sure, that in the range of accelerations for which the clock is supposed to be used, the clocks show the same time as momentary CMIO.
Implicitly, this implies that we assume such clocks can be made and that such definition makes good sense, i.e. it makes the theory simple.
Mathematically, this means that the clocks will show proper time, which motivates this postulate. In detail, in CMIO we can measure the curve on which the NIO moves through spacetime to be $\gamma(t),$ which we parametrize by our time $t$. The expression of the curve in Minkowski coordinates in some inertial reference frame is $$\gamma(t)=(t,x(t),y(t),z(t)).$$
The tangent vector to the curve, i.e. the four velocity of NIO at time $t$, is then $$\vec{v}=(1,\dot{x}(t),\dot{y}(t),\dot{z}(t)).$$
And we can use this to compute infinitesimal spacetime interval between two events on the curve separated by infinitesimal time $dt$ (which means the two events are separated by the vector $\vec{v}dt)$:
$$ds^2=\eta(\vec{v}dt,\vec{v}dt)=\left(-1+\dot{x}(t)+\dot{y}(t)+\dot{z}(t)\right)dt^2,$$
where $\eta$ is Minkowski metric. But if we pick CMIO which is momentary at rest with NIO at time $t$, then we have $ds^2=-dt^2$ at this time, since time derivatives of spatial coordinates are zero at time $t$. The spacetime interval of these two events is by the definition, the (negative squared) proper time that passed between these two events. So if we demand that the clocks in NIO show the same readings as CMIO clocks at time $t$, then they will show  the proper time, i.e. (square root negative)  spacetime interval.
Globally, the proper time passed between two events $A$ and $B$ on the curve is simply sum of infinitesimal proper times
$$\tau=\sum_i{d\tau_i}=\sum_i{\sqrt{-ds^2_i}}.$$
Since spacetime interval is invariant under Lorentz transformations, we can compute each $ds^2_i$ in different frame. If we pick at each time the relevant CMIO, the spacetime interval simplifies to $ds_i^2=-dt^2_i$, where each $dt_i$ is expressed in different frame. What this ammounts to is that
$$\tau=\sum_i{dt_i},$$
but this is exactly what our clocks are supposed to measure as demanded by the postulate.
So, the postulate basically tells us, that well made clocks wore by an observer moving on the timelike curve should show length of the curve $$d\tau=\int_\gamma{-\sqrt{\eta(\dot\gamma(t),\dot\gamma(t))}}dt$$
as motivated by purely geometrical considerations with no reference to physics whatsoever.
A: My understanding of the clock postulate is that observations of (not by) the two clocks reveals no difference between the clock of the NIO and the one of the CMIO.
The NIOs acceleration affects how he sees other clocks.  His own clock is unaffected.  Clocks in the direction of his acceleration appear to him to be running faster, those in the opposite direction appear to be running slower.
The effect is linear and proportional to their distance.
That is why the leading and trailing observers with two accelerating clocks agree which clock seems to be running faster than the other.
The CMIO does not see this effect.  Looking at any series of events in the direction of his acceleration, The NIO will see them proceeding at a faster rate than the CMIO sees them.  The accelerating third particle will seem to the NIO to be accelerating at a greater rate if it is in this direction.
If the acceleration of the NIO is away from the third particle he will see it accelerating more slowly than the CMIO sees it.
A: The point is controversial. Some claim it is an independent assumption while others that the effect of acceleration on proper time can be calculated.
In my opinion, those derivations are circular.
My view is that it must be formulated as an independent assumption and that it cannot be derived. So "the clock postulate" or the "clock hypothesis" correctly describes the effect.
It is not about how the clock mechanism is affected by acceleration. It is about how the fundamental laws of physics (including the passage of time) are affected by acceleration.
You cannot instantaneously apply a Lorentz transformation because Lorentz transformations are between "inertial systems". So the fact that we can extend them locally to not inertial frames is not a logical imposition but an additional assumption.
For the "shut up and calculate" approach this issue is irrelevant.
