First of all, can the color even be seen? James Webb has a spectrum of 600 nm at the lowest end, which means it can just barely see the color red. In addition, it could potentially be seen in other wavelengths that aren't visible.
The spatial resolution is around 70 milli-arc seconds. That means that the Roadster, being about 4m in size (roughly) in it's longest dimension could be resolved at about 11,800 km. That would require an exceedingly close pass! Of course, it can be seen even if not resolved, but it will still be quite dark.
Basically James Webb would treat a Tesla Roadster similar to a 4m asteroid. From this paper, it seems like even the much larger 275m sized asteroids would be difficult to see.
Bottom line, it might show up as an interesting speck, but is unlikely to actually be observed unless intentionally attempted at a particularly close approach.