Is saying "Bismillah" when placing leftover food somewhere, with the intention that jinn may eat from it, a correct act that has a basis in the Sunnah, or is uttering the Basmala at the beginning of eating sufficient?

There is no evidence for the legitimacy of saying "Bismillah" (in the name of Allah) when placing food scraps in the trash. The Sunnah mentions saying "Bismillah" at the beginning of eating, or if one forgets, saying: "Bismillah awwalahu wa akhirahu" (In the name of Allah, at its beginning and its end), and saying "Bismillah" over a container when it cannot be covered.

As for saying "Bismillah" over bones after finishing with them and discarding them for the Muslim jinn to benefit from, there are two views among scholars: The first is that bones from an animal slaughtered with the name of Allah mentioned over it can be benefited from by jinn just as humans benefit from them. The other view is to say "Bismillah" when discarding the bone so that the jinn may benefit from it.

Furthermore, if excess food cannot be preserved, given to the poor, or to animals, it should be placed in a clean spot, and it is legislated to say "Bismillah" over it so that the jinn may benefit from it, based on the Prophet's (peace be upon him) statement to the jinn: "For you is every bone over which the name of Allah has been mentioned..."