How to determine best fit from two?
It looks like a straight line is not a great approximation in either case. One thing which is unclear to me is that the y-axis in the top plot is $\sigma^2/v$ whereas in the second plot it is just $\sigma^2$ (as far as I can see), and not $\sigma^2/v^2$ as I expected.

What you need to figure out is what you should actually expect. When you say that $\sigma^2$ should depend on $v^2$ then this doesn't say much. From what you do it looks like that you expect those two quantities to be proportional:

$$\sigma^2=\alpha v^2$$

because then you would ideally have a constant ratio $\sigma^2/v^2$. Is this the case?

Note that if you model $\sigma^2/v$ as a linear function of $v$, then you have

$$\sigma^2/v=mv+q$$

which is equivalent to

$$\sigma^2=mv^2+qv$$

So you actually model $\sigma^2$ by a quadratic function of the variable $v$.

To determine the best fit, it seems natural to compare the normalized approximation errors of the different fits. Given that $d_i$ are your measured data points, and $v_i$ are the discrete velocities corresponding to the data $d_i$, the approximation error can be defined as

$$\epsilon=\frac{\sum_i \left[d_i-f(v_i)\right]^2}{\sum_id_i^2}$$

where $f(v)$ is the approximation function, which could be linear as you've suggested in your question, i.e. $f(v)=mv+q$.