There should be no issue with licensing the Kodak name for film only to an entity that might purchase the film division. Licensing your name for specific products only is done all the time. Kodak would retain the use of the name for their inkjet products and whatever else remains. Already Kodak chemistry is made under license by the owner of the former Kodak chemical plant.
A point to ponder is: When the sales of a product decline to the point that a master roll of film cannot be cut and sold out completely while still "in-date", then it becomes impractical to continue manufacture. Kodak can only make a certain size of master rolls, which are quite large, mile long I think, and wider than a yard (3-4 ft). What I don't understand is why they discontinued the 8x10 size of film that is still available in 4x5, as if there is stock for 4x5 there is stock for 8x10, as it is just a different cutting of the same master roll.