I used the National Park Service's "Civil War Soldiers and Sailors System" (Google soldier sailor system & it'll take you there), and a real quick check of the Mississippi Sons of Confederate Veterans version of Mississippi regimental histories. Note that both "Cook's Company" and "Bradford's Battery" (a.k.a. co. F First Mississippi) were both recruited in Lawrence Co. Mississippi, real close to Louisiana.
I feel like I'm on very thin ice here, for I'm merely speculating that there's some sort of connection, and I'd advise you to research this all much more closely before you conclude any of it is "truth". "Cook's Company" has NO unit history, per se, unless it is the same unit as Bradford's battery, but that hypothetical conclusion is going to require a great deal of research to confirm (and might not be confirmed). Separate fact from intuition / speculation!
I found little (or, truthfully, nothing) in the "Official Records" (on-line at Cornell's "Making of America" web site) on "Cook's Company" or any of several variations on such a name.