There are many papers which derive the form of the Lorentz transform from elementary symmetry principles (usually homogeneity of spacetime, isotropy of space, and the fact that boosts form a group), e.g. here or here (warning: PDF). These derivations result in an expression for the Lorentz factor of the form $$\gamma(v) = \frac{1}{\sqrt{1 - K v^2}},$$ where $K$ is some non-negative constant. When $K$ is strictly positive, the Lorentz transform is recovered, whereas when $K$ is zero, the Galilean transform is recovered. This style of derivation implies that the Galilean transform and the Lorentz transform occupy similar roles in non-relativistic and relativistic physics, respectively.
However, in relativity, boosts between reference frames (which are described by the Lorentz transform) are part of the isometry group of the Minkowski metric. Galilean boosts, on the other hand, are never (to my knowledge) presented as isometries of the Euclidean metric; in fact, I'm having difficulty showing that a Galilean boost is an isometry.
The other option is that Galilean relativity (i.e. invariance under Galilean boosts) is actually a dynamical symmetry (i.e. a symmetry of the equations of motion, specifically the Euler-Lagrange equations), and not an isometry. Is this the case, and if so, can invariance under Lorentz boosts be similarly conceived as a dynamical symmetry (of the E-L equations for a relativistic field, perhaps)?