Looking for more accurate energy-momentum transformations for photons

[SiennaTheGr8]

hi, after using Ibix/Sienna's method, we get $f'=γ(1-βcosθ)f$, which is supposed to match the frequency/energy relations right?

Well, that is the frequency-transformation formula you obtain, but if you want to use it to show that light's energy transforms the same way then you need to invoke the Planck–Einstein relation for photons. It's "cheating" in a sense, and not entirely satisfying, but it's not "wrong," and it's not entirely ahistorical actually (Einstein's paper on the photoelectric effect was published a bit before his paper on special relativity, though it would take several years before the Planck–Einstein relation was widely accepted).

It's a bit trickier to show that a classical light wave's energy transforms in that way. You need to transform the appropriate electromagnetic energy density ("appropriate" means that $\mathbf{E} \perp \mathbf{B}$ and $E = B$), and then you need to multiply the result by the transformation of the appropriate volume (what "appropriate" means here is rather subtle, and I'm still grappling with what Einstein's doing there at the beginning of §8 of his SR paper).