My idea is to write a story about the band, that I had heard might or might not be true. In order to write it in a fictitious fashion, I would have to create a whole new band, and even deal with having to write original music. That is not my thing. So, I want to use the real band, and write from there.
The Rolling Stones are considered to be public figures and so the standard of protection afforded them is somewhat different from that afforded to private citizens.
That being said, no one could use so much as five seconds of their music without their permission (or, at any rate, the permission of whoever currently holds the rights -- I don't know if it's them or not).
And I don't see how you could possibly write a story about the Rolling Stones without using their music in the story.
That means that I don't see how you could write a movie about them without their permission and active cooperation -- and their agreeing to let you use their music.
And of course, they don't really "give" anyone permission to use their music. You, or whomever, has to pay for it. Pay a lot for it. A fortune.
So however you proceed, it would certainly present significant problems.
If you were already a major established writer/director you might approach them with the proposal. But you're not and so you can't (unless by some chance you happen to be related to or know or know somebody who's pals with one of the Rolling Stones). And even if you were to write the script on a hope, a song, and a prayer, there is no practicable way of getting it to them in an official capacity. And without getting it to them, and getting them to give you their permission and active cooperation -- in writing -- the project would be dead in the water. You simply wouldn't be able to get in the door with it anywhere.
So fictionalizing may be the only practical approach, however much of a compromise it may be.
NMS