Almost :-). The most simple way to describe special relativistic point mechanics is to use a Lagrangian of the form
$$L=\frac{m}{2} \dot{x}^{\mu} \dot{x}_{\mu} + L_{\text{int}}(x^{\mu},\dot{x}^{\mu}),$$
where the dot denotes derivatives with respect to proper time and ##L_{\text{int}}## is homogeneous of degree 1 wrt. ##\dot{x}^{\mu}##, i.e.,
$$\dot{x}^{\mu} \frac{\partial L_{\text{int}}}{\partial \dot{x}^{\mu}}=L_{\text{int}}.$$
Then ##\tau## is an affine parameter, i.e., since ##L## is not dependent explicitly on ##\tau##, the quantity
$$H=p_{\mu} \dot{x}^{\mu}-L=\frac{m}{2} \dot{x}^{\mu} \dot{x}_{\mu}=\text{const},$$
and thus you can choose ##\tau## as the proper time, so that ##H=mc^2/2##.
The most simple equation of motion, fulfilling this properties, is provided by
$$L_{\int}=\frac{q}{c} A_{\mu} \dot{x}^{\mu},$$
where ##A_{\mu}=A_{\mu}(x)## is a vector field and ##q## a parameter. Then the Euler-Lagrange equations give
$$p_{\mu}=m \dot{q}_{\mu}+\frac{q}{c} A_{\mu},\\
\dot{p}_{\mu} = m \ddot{q}_{\mu} +\frac{q}{c} \dot{u}^{\nu} \partial_{\nu} A_{\mu}=\frac{q}{c} \dot{q}^{\nu} \partial_{\mu} A_{\nu}$$
or
$$m \ddot{q}_{\mu} = \frac{q}{c} F_{\mu \nu} \dot{x}^{\nu}, \quad F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu},$$
and that's the relativistic equation of motion for a particle in the electromagnetic field, represented by the four-potential ##A_{\mu}##.
It is clear that only ##F_{\mu \nu}## is observable, and that indeed the equations of motion do not change under the gauge transformation
$$A_{\mu}'=A_{\mu}+\partial_{\mu} \chi,$$
where ##\chi## is an arbitrary scalar field. Indeed the change in the Lagrangian is a total derivative of a function of ##x## only:
$$L_{\text{int}}'=L_{\text{int}} + \frac{q}{c} \dot{x}^{\mu} \partial_{\mu} \chi=L_{\text{int}} + \frac{q}{c}+\frac{\mathrm{d}}{\mathrm{d} \tau} \chi.$$
The equation of motion is thus gauge invariant.
