Bioshock is a popular franchise and Big Daddy could bring a different playstyle, being a heavyweight with movement options. That said, I don't like your special moves - I say take a page out of PSASBR's book.
Neutral Special: Flaming Plasmid
Big Daddy charges up some fire in his hand, then shoves it forward into a flaming burst in front of him. Think of it like a larger-scale version of Mario's forward smash.
Side Special: Charging Rush
Basically your idea. Big Daddy charges forward (or in the air, diagonally downward at a 45 degree angle). This move can be cancelled early with a normal attack or a jump, giving him a lot of mobility options to rush down opponents. This attack also grants Big Daddy super armor, making it difficult to stop.
Down Special: Ice Plasmid
Big Daddy drops some cold liquid from his hand onto the ground, which causes a puddle to form on the ground for about 10 seconds. Any opponent who steps in it is frozen temporarily.
Up Special: Air Plasmid
Big Daddy forms a tornado under himself to rise him into the air. He can dismount at any time with an attack, but he's left helpless afterwards.
I was considering Lightning Plasmid, but I felt like it wasn't in his playstyle to use projectiles - he'd rather just charge at his opponents and beat them down. For his normal moves, we can include some generic punches in his tilts while including the drill into his smashes (and potentially some aerial/throw moves).
Also, not sure how hard this would be to code... but it'd be a neat aesthetic thing for his eyes to change color based on his distance to an opponent.
... also, I've never played Bioshock, but hopefully this sounds good to someone who has.