Ok so I solved it.
I was under the impression that when you enabled bUsePawnControlRotation on the spring arm, it inherits the actual rotation of the actor it is attached to. I should have looked more closely at the name of the actual variable though, because it does in fact inherit the control rotation of the pawn, which is updated by AddControllerPitchInput() as @Tuerer suggested I use.
Feeling kinda stupid right now, but apparently I don’t have to do anything other than enabling bUsePawnControlRotation, bInheritPitch, and bInheritYaw.