A piece and sausage or a roll and sausage (hello regional terminology what are you doing here) are distinct things from a hot dog, despite them all being sausage with bread - so I would be less inclined to call it a sandwich, while I might call the first two a sandwich.

A hot dog has connotations in my head. Perhaps the type of sausage is what assigns the title? Regular links sausage in a finger roll still isn't a hot dog despite it being the exact same.

Ponderous.