For if I started by trying to make the Hamiltonian Lorentz invariant, I would have failed. Indeed, the Hamiltonian is part of a covariant tensor. But how do I know that the Lagrangian is not a part of such a tensor?
All of Lagrangian formulation, Hamiltonian formulation, and Poisson bracket formulation involve some partial derivative with respect to time at some point. None of them are manifestly Lorentz invariant.
We knew the system governed by Klein-Gordon equation is Lorentz invariant, but could I construct a non-Lorentz-invariant Lagrangian and derive a Lorentz invariant equation of motion out of that?