Felix,
For reasons I mentioned, I wouldn't advocate the use of transformations. We might also consider that fields transform differently than particles. It could be argued that the Galilean transformation is suitable for a particle in the case where no change in momentum occurs. But of course, as soon as the particle starts interacting with a field the usefulness of the Galilean transformation dissolves.
Field or wave equations are vastly superior and more rigorous than field transformations in my opinion and can be derived very easily from the results of my paper. So I'd argue that there is no single transformation, modification of the Lorentz transformation or otherwise, that rigorously applies to both particles and fields. And that is especially vivid when a transfer of momentum or energy occurs.
However, if you still wish to consider possible modifications of the LT, we could leverage these findings:
[math]\gamma = \frac{1}{\sqrt{\frac{v^2}{c^2}}} = \frac{\omega}{\omega_p} = \frac{\hbar \omega}{\hbar \omega_p} = \frac{1}{\hbar \omega_p} E[/math]
[math]\gamma = \frac{1}{\sqrt{\frac{v^2}{c^2}}} = \frac{\omega}{\omega_p} = \sqrt{1 \frac{c^2 k^2}{\omega^2}} = \sqrt{1 \frac{\hbar^2 c^2 k^2}{\hbar^2 \omega^2}} = \sqrt{1 \frac{1}{\hbar^2 \omega^2} \mathbf{p}}[/math]
[math]\frac{\mathbf{v}}{c^2} = \frac{1}{\mathbf{v_{phase}}}[/math]
