The "Taylor Hypothesis" is founded on the idea that the changes observed in any given plasma measured in the solar wind propagate at speeds much much less than the bulk flow speed of the solar wind (well, this has been applied in other regions of space than just the solar wind, but it's most commonly assumed there). We start with the nonrelativistic Doppler relation (i.e., assume V/c $\ll$ 1) given by:
$$
\omega_{sc} = \omega + \mathbf{k} \cdot \mathbf{V} \tag{0}
$$
where $\omega_{sc}$ is the observed angular frequency in the spacecraft frame, $\omega$ is the intrinsic angular frequency in plasma frame moving at $\mathbf{V}$, and $\mathbf{k}$ is the wave vector. Note that for V/c $\ll$ 1, the wavenumber and wave unit vector are invariant under transformation (well, they are within any realistic application).
Note that in general, one measures $f_{sc} = \tfrac{ \omega_{sc} }{ 2 \pi }$, not directly $\omega_{sc}$. In any case, Equation 0 can be rewritten as:
$$
\frac{ \omega_{sc} }{ k } = \frac{ \omega }{ k } + \frac{ \mathbf{k} \cdot \mathbf{V} }{ k } \tag{1}
$$
where $\tfrac{ \omega }{ k }$ is the rest frame phase speed of the fluctuation. Note that although some refer to a "phase velocity" it is not a true 3-vector velocity as it does not transform like any other Galilean or Lorentz 3-vector would. This is because the vector direction is controlled by $\mathbf{k}$, which would be in the denominator. In contrast, the group velocity does properly transform like other 3-vector velocities.
So back to the question at hand. The Taylor Hypothesis relies on the following assumption $\tfrac{ \omega }{ k } \ll \tfrac{ \mathbf{k} \cdot \mathbf{V} }{ k }$, in which case all time-variations observed in the spacecraft frame should be the result of spatial variations in the plasma rest frame. It's a fancy way of saying that you observe changes in your data because different things are moving past your spacecraft.
Then if we wanted to convert the spectrum to the spatial domain could we convert frequency to wavenumber...
The wavenumber is an angular quantity so yes, there needs to be a $2 \pi$ term (i.e., your 2nd expression). In regards to an FFT, calculate the FFT frequencies [Hz] then convert those to angular frequencies [rad/s], then divide by the magnitude of the bulk flow velocity, $V$. This would provide you with a temporal-to-spatial scale conversion assuming the fluctuations ONLY propagate along the solar wind bulk flow velocity direction.
There are fancier ways to handle this where one need not make such assumptions, but they suffer from the same issues that the simplistic approach above does. That is, these approximations are really only valid for the largest(lowest) scales(frequencies) because even at ~0.1 Hz (in the solar wind, near-Earth environment) measured in the spacecraft frame one can have fast/magnetosonic waves, which have phase speeds comparable to the solar wind speed. These violate the Taylor Hypothesis and are well known to be common in the solar wind.
