Sunshine, perhaps? https://en.wikipedia.org/wiki/Sunshine_(2007_film)
Brilliant, brilliant film. The premise is shonky as they come but it's obvious that the film itself has little to do with the premise. Awesome acting, literally-awesome SFX (their ability to give the sun so much weight is amazing, you really get a sense of an immortal unstoppable force), and a message that's neither shallow nor deep but prompts some self-reflection in ways you don't expect.
To be more clear, I don't think the creators were thinking "will this scene make us more money", nor do I think it's their job. And on the topic of realism in a movie, unless it's a historical film or the goal is to educate, then realism should be near the bottom of the checklist when creating a scene.
It does, though not being a rigid body, the rate of rotation is a function of latitude. Near the equator, the period of rotation is ~24 days. As you near the poles, the period is closer to 30-35 days. A means of measuring these rates available to backyard astronomers is through measuring the positions of sunspots over time. [1]
[1] Please don't look at the sun through a telescope or really at all without knowing what you're doing and taking appropriate precautions: https://www.youtube.com/watch?v=R9cMXCemoJI
Here's a really old picture that is at essentially the same scale as what the BBC decided to show, but made in 1997: https://en.wikipedia.org/wiki/Swedish_Vacuum_Solar_Telescope...