The short answer is that yes the color would have to change somewhat. All the stars in the instability strip (RR Lyrae, Cepheids, etc) are variable because they pulsate. Over the course of the pulsation, they vary in brightness, size and surface temperature.
It is the temperature change that is important here. As the temperature goes up the luminosity increases for a given amount of surface area on the star. This plus the changing size account for variations in brightness. However, as the temperature changes, the peak wavelength changes as well (as described by Wein's Law). This means that the star gets redder and bluer over the course of the pulsation cycle.
Now as to the amount of change and how measurable it would be, that I'm not sure of. At a fundamental level there has to be a change but whether it is noticable with broadband (or even narrowband) photometry, is a different question. I know people who might know (as this is what they study) and I'll ask them and see if I can find a reference as well.