Your second equation is incorrect. In both Newtonian Mechanics and Relativistic Mechanics, force is the time rate of change of momentum:
$\vec F = \dfrac{d \vec p}{dt}$
Where, in special relativity, momentum is:
$\vec p = m \dfrac{\vec v}{\sqrt{1 - \frac{v^2}{c^2}}} = m \gamma \vec v$
and $m$ is the invariant mass.
Thus, when the time derivative is taken, by the product rule:
$\vec F = m(\gamma \vec a + \dot \gamma \vec v)$
Note that, in general, the force vector is not parallel with the acceleration vector!
If the force vector is always parallel with the velocity vector, the force equation simplifies to:
$F = m \gamma^3 a = m \dfrac{a}{(1 - \frac{v^2}{c^2})^{\frac{3}{2}}}$
Now, when you write:
It is moving with an uniform acceleration
you must actually be more specific. It appears that you're thinking about coordinate acceleration however, there is also the acceleration as measured by accelerometers (the proper acceleration) and this distinction is often not appreciated by SR "newbies".
While it is possible for the proper acceleration to be uniform, it isn't possible for the coordinate acceleration to be uniform as that would require unlimited force.
That motion was relative was realized by Galileo, so there was a theory of relativity -- Galilean relativity -- long before Einstein. That the speed of light should be the same according to all observers is indeed inconsistent with the Galilean relativity. This is because in Galilean relativity time is absolute. But it is not mathematically inconsistent to have a universal speed of light. We "just" have to give up the idea that time is absolute. (That's a very big "just" - a Nobel prize "just"!) From On the Electrodynamics of Moving Bodies:
Examples of this sort, together with the unsuccessful attempts to discover
any motion of the earth relatively to the “light medium,” suggest that the
phenomena of electrodynamics as well as of mechanics possess no properties
corresponding to the idea of absolute rest. They suggest rather that ... the same laws of
electrodynamics and optics will be valid for all frames of reference for which the
equations of mechanics hold good. We will raise this conjecture (the purport
of which will hereafter be called the “Principle of Relativity”) to the status
of a postulate, and also introduce another postulate, which is only apparently
irreconcilable with the former, namely, that light is always propagated in empty
space with a definite velocity $c$ which is independent of the state of motion of the emitting body.
The Galilean relativity says that if Alice moves at a velocity $v$ relative to Bob, their space and time coordinates are related by \begin{align}
t_A & = t_B \\
x_A & = x_B + vt_B
\end{align}
and it follows from this that if Alice emits a signal with velocity $w$, Bob will observe its velocity to be $v+w$. This is why Einstein says that the postulates are apparently irreconcilable, as you have found.
But in the Einsteinian relativity, Alice's and Bob's space and time coordinates are instead related by \begin{align}
t_A & = \gamma(t_B - v x_B/c^2) \\
x_A & = \gamma(x_B + \frac{v}{c} t_B)
\end{align}
where $$\gamma = 1/\sqrt{1 - \frac{v^2}{c^2}}$$ and $c$ is some constant with the units of velocity -- at this point we haven't related $c$ to physical phenomena like light yet. Now it turns out that when space and time are related like this, because Alice does not agree with Bob what time is, velocities add differently. If Alice emits a signal with velocity $w$, the velocity measured by Bob is $$v\oplus w = \frac{v + w}{1 + vw/c^2}.$$
Now there are two interesting things about this formula. First of all, if $v, w < c$ then $v\oplus w$, and -- as the resolution to the conundrum -- if $w = c$, then regardless of $v$, $v\oplus w = c$. So under this theory of relativity, signals moving at $c$ are special: everyone will agree that they move at $c$.
To relate this to light, one of Maxwell's equations -- the equations that govern electricity and magnetism -- is $$\nabla\times\mathbf B = \mu_0 \mathbf J + C^{-2} \frac{\partial \mathbf E}{\partial t}$$
where $C$ is a constant with units of velocity -- it is the speed of electromagnetic waves, that is, light. It turns out that Maxwell's equations are never consistent with Galilean relativity, but they are consistent with Einsteinian relativity, under the condition that $c = C$. So the $c$ that enters into the coordinate transformation is indeed the speed of light. (You can do the math to make absolutely certain that it is like this, but you can also think of it like this: under Einsteinian relativity $c$ is the only speed that everyone agrees on. So if everyone is to agree on the form of the equation, the only velocity that can show up is $c$.)
Best Answer
A synchronisation procedure of ideal clocks at mutual rest must be transitive, symmetric, reflexive and it must remain valid in time once one has adjusted the clocks to impose it. There is no evident a priori reasons why Einstein's procedure should satisfy these constraints. The fact that it instead happens is the physical content of both postulates you quoted.
Actually there is a third physical constraint: The value of the velocity of light must be constantly $c$ when measured along a closed path. This measurement does not need a synchronisation procedure, since just one clock is exploited.
A natural issue show up at this juncture: whether there are synchronisation procedures different from Einstein's one which however fulfill all requirements.
The answer is positive (without imposing other constraints like isotropy and homogeneity) and they give rise to other formulations of special relativity, which are physically equivalent to Einstein's one. (Geometrically speaking, it turns out that the geometry of the rest spaces is not induced by the metric of the spacetime by means of the standard procedure of induction of a metric on a submanifold.)
There are well-known physical situations regarding clocks with non-inertial motion (at rest with respect to each other), where Einstein's procedure cannot be used and other synchronisation procedures must be adopted. The most relevant is the one regarding a rotating platform. If I remember well, the first correct analysis of the problem was proposed by Born.