Fourier transform of moving image sensor
I feel that the question needs a little bit more clarifying but there probably is enough material here to provide some sort of response and adjust it later if needed. 

To cut a long story short: Rolling Shutter distortion. This concept is key here (irrespectively of whether or not you have a rolling or global shutter camera that is unsynchronised to the interferometer).

Now, I will treat $v, R$ as independent from time here for a minute. Also, $F_{FR}$ will stand for frame rate.

Your "problem" seems to be that due to $v$, each pixel row might actually be sampling spectral lines from two radically different wavelengths. If this is true then each one of your frames will acquire a slant whose angle will be proportional to $\frac{v}{F_{FR}}$. 

In the time dependent version, the frames are capturing a "curve" whose curvature is proportional to $\frac{v(t)}{F_{FR}}$ and...good luck with that.

Since you are dealing with interferometry, you are only really imaging ONE single focal plane, therefore, you could apply a simple (anti)-skew transform to counteract the skew caused by the rolling shutter. This, of course, is within limits. In any case, the rolling shutter effect of the camera inserts a phase error in the image which, to an extent, can be counteracted but in the end it all comes down to controlling $v$.

Assuming that your camera can maintain a constant $F_{FR}$ for the duration of the scan, then essentially, you don't have a 2D camera. 

I would like us to imagine it as one long linear $M \times N$ camera (because of $v$ and interferometry) only it is wrapped around with a "wrap" factor that depends on $v$. 

So, theoretically, you could find a $v$ which scans a spectrum range in one frame. Maybe $v$ is such that the spectral range is contained in two frames or three frames or four...Obviously, the more, the better. Therefore, each pixel row now has a $\lambda_{start}$ and a $\lambda_{end}$. So, I want you to imagine a chirp that is wrapped across the camera sensor or in fact across many different frames. 

Therefore, you could convert your single $M \times N$ image, into one long vector and then do a spectrogram on this vector. This vector is essentially encoding wavelength to space (across the camera sensor). 

Does this have any distortions? Of course it has, but it would be like trying to obtain the frequency response of a system by feeding it a chirp. Within limits it is perfectly possible. If you scan it too fast, you might skip a spectral line

Perhaps it is easier to see why $v(t)$ would insert a phase error to the "chirp" when considering the camera as one long image sensor (?).

Hope this helps.