How do I get the change in angular diameter value? Given that I got the equation relating the angular diameter(θ) and $V$–$K$ band value, which is known as $\logθ_0=(0.262±0.004)(V−K)_0+(0.547±0.006)$ from this paper, how do I get the 'change' in angular diameter value from that equation?

1) Can I just get the max and mix value of the $V$–$K$ value and insert each into the equation and get max θ and min θ and get the difference?

2) Or I need to the whole lists of angular diameter value over each period and insert the lists of value into the equation and plot the graph and then get the difference?

3) Or is there any other method to do this?
You can see from the paper you linked that they followed the procedure you outlined in option (2). Figure 4 of that paper shows the $\theta (t)$ for a few stars. They specifically state towards the end of section 6


  Figure 4 shows in the top panel the relation between the angular diameter predicted by the SB-relation versus radial variation, and in the bottom panel the angular diameter as a function of phase.


The SB-relation they're talking about is the one you cite in your question. So your process should be to take your list of $V-K$ colors over time and convert them to angular size using the surface brightness relation, then fit a function to that (via cubic spline, least squares, etc.) and determine the total angular variation over time.

I will note also that the $\theta$ vs $V-K$ relation you've cited here is equation 10 in the linked paper which is not a result of that paper. Their result is listed in equation 9 with equations 10 and 11 being results from other, previous work for the reader to compare. You should sure of which equation you want to use and cite the appropriate source.