Deriving the action and the Lagrangian for a free massive point particle in Special Relativity
Yes, the invariance of the action follows from special relativity – and special relativity is right (not only) because it is experimentally verified. All the equations of motion may be derived from the condition $\delta S = 0$, the action is stationary (which usually means it has the minimum value on the allowed trajectory/history among all trajectories/histories with the same initial and final conditions). If $S$ depended on the inertial system, so would the terms in the equations $\delta S =0$, and these laws of motion couldn't be Lorentz-covariant (note how this Lorentz is spelled; Lorenz also existed but it was a different physicist). Quite generally, you shouldn't think about "derivation of the action". When we work with the action at all, we are doing so because we view the action as the most fundamental expression – and we derive everything else out of it. In that context, we pretty much define a Lorentz-invariant theory as a theory determined by a Lorentz-invariant action.
Your integral is Lorentz-invariant but it is not translationally invariant under $x^\mu \to x^\mu + a^\mu$. So it's not Poincaré-invariant (the Poincaré symmetry unifies the Lorentz transformations and spacetime translations) and due to this violation, we also say that it disagrees with the laws of special relativity. You could also create other expressions, e.g. replace $x_\mu x^\mu$ in the integral by some extrinsic curvature invariant of the world line etc. Those terms could be made Poincaré-invariant. So the right claim is that the proper length of the world line is the only Poincaré-invariant functional that doesn't depend on any higher derivatives of the coordinates $x^\mu(\tau)$.
In SR the equations have the same form in all inertial frames. This implies that the action of a free particle is that shown by L&L.
For convenience, let us take the proper time $s$ of the massive particle as the integration variable in the action, with a Lagrangian given by $L=L(x^\mu,\dot{x}^\mu,s)$, where $\dot{x}^\mu=dx^\mu/ds$. We have to prove that $L=-a$ (constant).
The Lagrangian formalism implies that, by changing the coordinates $x^\mu \rightarrow x'^\mu$, the equations of motion obtained from the Lagrangian $L=L'(x'^\mu,\dot{x}'^\mu,s)$, regarded as a new function of the new coordinates, are equivalent to the equations of motion (the Euler-Lagrange equations) for the old coordinates.
In the case of a Poincarè transformation of coordinates, the new equations describe the motion in a new frame of reference and SR requires that they have the same form as the old equations. As usual in the Lagrangian formalism (as explained in L&L's Mechanics), this is possible if $L'(x'^\mu,\dot{x}'^\mu,s)$ differs from $L(x'^\mu,\dot{x}'^\mu,s)$ by the derivative of a function of the coordinates and the proper time,
$$L(x^\mu,\dot{x}^\mu,s)=L(x'^\mu,\dot{x}'^\mu,s)+\dot{F}'(x'^\mu,s).$$
It is apparent in this equation that in the old frame of reference $\dot{F}(x^\mu,s)=0$. As the old inertial coordinates are not special, SR requires that $\dot{F}'(x'^\mu,s)=0$ in the new (arbitrary) inertial frame. Thus, $L(x^\mu,\dot{x}^\mu,s)=L(x'^\mu,\dot{x}'^\mu,s)$, which means that the Lagrangian and the action are invariant.
As this equation is valid for any constant space-time translation $x^\mu \rightarrow x'^\mu=x^\mu+a^\mu$, $L$ does not depend on $x^\mu$. L&L's argument implies that $L$ does not depend on $\dot{x}^\mu$ (since $\dot{x}_\mu\dot{x}^\mu=-1$ is the only scalar that one can construct). Finally, $L$ does not depend on $s$ because the time is uniform for a free particle. Therefore, $L=-a$ is a constant.
Notwithstanding, note that L&L shows in their book that $\dot{F}(x^\mu)$ does appear in the Lagrangian of a charged particle in the absence of the electric field and the magnetic field (a free particle in the classical sense),
$$ \dot{F}(x^\mu)= \frac{e}{c}A_\mu\dot{x}^\mu, $$
where the 4-vector potential $A_\mu$ is a "pure gauge",
$$ A_\mu=\frac{\partial \chi}{\partial x^\mu} $$
for a scalar $\chi$.
In fact, gauge transformations introduce $\dot{F}$ in their Lagrangian of the charged particle interacting with the EM field without changing the equations of motion.