The defining property of inertial forces is that they don't affect Galilean invariants: distance and duration.
This requires that they be proportional to mass, but that doesn't fully characterise them.
The invariant definition can probably be proven to be equivalent to the more intuitive but arguably less fundamental definition, that a time-dependent SO(3) transformation can remove them entirely.
We can imagine some Galilean spacetime coordinate systems which introduce forces that might be described as "hyperinertial". These would violate the integrity of rigid bodies -- there's probably some connection to SO(3) here as well, what with rigid bodies being determined by an element of SO (3) and a translation from the origin.
I think Lorentzian relativity can't have inertial forces the way Galilean relativity does. Invariance of the spacetime interval fixes the Poincaré group (Lorentz group if origin fixed) completely all on its own, whereas distance and duration doesn't fix the Galilean group. You need to impose linearity on top. This must have something to do with the fact that Galilean symmetry decomposes directly between space and time, whereas Lorentzian only semidirectly.