The total energy of an object comes from the time part of the four-momentum, and so isn't a Lorentz invariant. On the other hand, is the potential energy of a compressed spring a Lorentz invariant?
The potential energy is not a Lorentz invariant, it is just a field energy for the spring parts, including the most significant contribution, the increased electronic energy from the compression of the electronic wavefunction, and it transforms along with the field momentum as a four-vector. In general, it isn't useful to separate energy into potential energy and kinetic energy in special relativity, the energy must have local flow, and is described by field energy, a stress-energy tensor, and it is due to a combination of local fields.
It depends of what potential energy you are using. If you refer to a potential energy that depends only on position variables, $V=V(x)$, this is not Lorentz invariant. If you refer to potential energy that depends both on position and time variables, this can be Lorentz invariant. Some examples of Lorentz invariant potential energies are given in the textbook by Schieve 
 Classical Relativistic Many-Body Dynamics; M. A. Trump and W. C. Schieve; 1999; Kluwer Academic Publishers