Display a rotating vector on four LEDs
Well, you've got the core thing correct:

Perception is non-linear. For \$n\cdot 90° + 45°,\,n\in\mathbb N\$, yes, the brightness of the adjacent LEDs should be identical, but that's about it.

Yes, we generally model perceived brightness to be logarithmic to irradiation. That doesn't mean the base of that logarithm is known for any color, for any human.

It also isn't clear that at the very low end of your PWM duty cycle, this will really be a sufficient model; so, I'm afraid, you'll have to do some characterization yourself, if you want to do this "properly".

Luckily, I think that if you're aiming for direction visualization with four LEDs of which you at most use 2 at the same time, you won't lose much "perceived accuracy" if you just try out one or two exponents, and settle for one that you deem "good enough". No magic involved, just try it out.