To elaborate on Mark M's answer:
If you consider an accelerating reference frame with respect to Rindler coordinates (where time is measured by idealized point-particle accelerating clocks, and objects at different locations accelerate at different rates in order to preserve proper lengths in the momentarily comoving reference frames), then light may not move at c, and can in fact even stop.
Specifically, for motion in one dimension, consider the transformations in natural units ($c=1$) between cartesian co-ordinates $(t,x)$ to Rindler co-ordinates $(t_R, x_R)$, for an observer accelerating at a rate of $g$ from an initial position $x_I = 1$, in order to maintain a fixed interval from the origin:
$$\begin{align*}
t_R &= \tfrac{1}{g}\mathop{\mathrm{arctanh}}\left(\frac{t}{x}\right) \;,
&
x_R &= \sqrt{x^2 - t^2\,}\;;
\tag{C $\to$ R}
\\[2ex]
t &= x_R \sinh(gt_R) \;,
&
x &= x_R \cosh(gt_R) \;.
\tag{R $\to$ C}
\end{align*}$$
A light signal emitted from some initial position $x_\varphi$ along the X-axis follows the trajectory $x = x_\varphi + vt$, where $v = \pm 1$ just gives the direction. Consider the trajectory that it follows in Rindler co-ordinates:
$$\begin{align*}
x_R^2 = x^2 - t^2 &= (x_\varphi + vt)^2 - t^2
\\
&= x_\varphi^2 + 2x_\varphi vt \tag{as $v^2t^2 - t^2 = 0$}
\\
&= x_\varphi^2 + 2x_\varphi vx_r \sinh(gt_R)\;;
\end{align*}$$
using the quadratic formula, we obtain
$$\begin{align*}
x_R &=
x_\varphi\Bigl[v\,\sinh(gt_R) + \cosh(gt_R)\Bigr]
\;=\; x_\varphi\exp(\pm gt_R) \;, \quad\text{for $v = \pm 1$}.
\end{align*}$$
Yes, that's an exponential function on the right.
It follows that the speed of a light signal is dependent on position in Rindler co-ordinates: the speed of the light signal emitted at $t = 0$ at $x_\varphi$ is
$$ \frac{\mathrm dx_R}{\mathrm dt_R} = \pm gx_\varphi \exp(\pm g \cdot 0) = \pm g x_\varphi \;.$$
We can show that the speed of light is a function only of position as follows.
A light emission (to either the left or right) from $x_1 > 0$ at $t_1 = 0$ reaches a position $x_2 = x_1 \exp(\pm g t_2)$ after an elapsed time of $t_2$; its speed at that time is $v_{1\to2} = g x_1 \exp(\pm g t_2)$, which is equal to the instantaneous speed of a light signal sent from the position $x_2$ at time $t_1 = 0$. So in natural units, the speed of light in Rindler co-ordinates is
$$ c(x) = gx \quad\Bigl[\text{in non-natural units,}\;\; gx/c\Bigr],$$
where $x$ is the location of the light signal.
In particular, any light signal appears to travel at the inertial constant speed $c$ just as it passes them.
This has a few consequences. Light signals sent from positions $0 < x_\varphi < x_I = 1$ will move more slowly in the proper time of the Rindler observer, with light signals moving to the right taking longer than usual to catch up to the accelerating observer, up until it reaches them, at which point it seems to travel at $c$. As we take $x_\varphi \to 0$, light signals in any direction appear to slow to a stop. Such beams of light define the Rindler horizon of the reference frame, cutting away a region of space-time from which the observer cannot obtain any information because they see the objects in it accelerating away too quickly, as with the event horizon of a black hole. Conversely, light signals at positions $x_\varphi > x_I = 1$ may appear to be travelling faster than c.
Best Answer
Almost. The approach you speak of is essentially Ignatowski's 1911 Approach. He used the relativity postulate (your assumption 1) but replaced the constant lightspeed assumption with some very basic assumptions about symmetries in spacetime (below). I also cite some more modern references for the approach.
This approach deduces the existence of a speed $c$ that would be observed to be the same in all inertial reference frames. The approach does admit $c=\infty$, i.e. Galilean relativity is admitted by this approach. Moreover, the approach cannot tell you what the invariant speed is if it turns out to be finite. It then becomes a wholly experimental question what this frame invariant speed is, or even whether it is finite. Of course, the Michelson Morley experiment and others showed that the invariant speed is finite, and that the speed of light is either that speed or mighty near to it. Whether the two are indeed the same boils down to the photon mass; they must be if the photon is truly massless and modern measurements give fantastically small upper limits to the photon mass. See the Photon Mass Wikipedia page for further details.
In this way, the universal invariant speed becomes a great deal more basic notion, a geometrical notion that is much deeper than light alone. Experiment then links the notion with the speed of light.
Note that, as in Pentcho Valev's answer, frame invariance of $c$ cannot be deduced from Maxwell's equations without further assumptions. It was simply assumed in the nineteenth century that Maxwell's equations would change their form to reflect the existence of a luminiferous aether until the Michelson Morley experiment suggested that they might not change their form.
The Assumptions
The assumptions you need to derive the Lorentz transformation by Ignatowski's approach are:
First, one must postulate that a manifold structure / co-ordinates on spacetime are even meaningful, and then that motion can be described by transformations on these co-ordinates.
The relativity postulate then ensures that these transformations depend only on the relative velocity between inertial frames and also completes the group structure for the set of transformations kitted with composition (through enforcing associativity).
Assumptions of homogeneity of spacetime together with continuity of the transformations in their dependence on the spacetime co-ordinates then show that the group of transformations is a linear, matrix group;
Assumption of continuity of the transformations in their dependence on the relative velocity between frames then shows that this matrix group is a Lie group of $4\times4$ matrices;
Isotropy of space then shows that the Lie group is the identity connected component of one of the orthogonal groups $O^+(4)$ or $O^+(1,\,3)$ (or the Galilee group, in the special case where the free parameter $c$ is infinite);
Causality then rules out the rotations $O^+(4)$;
Experiment shows us that the free parameter $c$ is finite and our group is $O^+(1,\,3)$, not the Galilee group.
References
Palash B. Pal, "Nothing but Relativity," Eur.J.Phys.24:315-319,2003
Jean-Marc Levy-Leblond, "One more derivation of the Lorentz transformation"