Haven't tried cotton in a Babolat but have in a Prince Shark, Volkl V1 and Yonex RDX 300. My method is to stuff cotton as far up the handle as possible using something like a pencil. I then keep packing it in and after the first few cotton balls I can assure you that they don't move. I keep stuffing and packing tight with the pencil and eventually the space in the handle fills up to the handle butt.
Using this method the cotton balls don't get past the handle, most likely because I use as large as will fit clumps and as it is not liquid I could not see it getting into the throat or head. As the cotton is packed in tightly and compressed with a pencil as each clump is added, uniformity is pretty good.
I just went back to my Prince Shark & took the cap butt handle cap (trap door) off to double check, and after 2 months the cotton is still packed in as tightly as it was on day one on both sides of the racquet handle channel.