Super Smash Bros. Brawl: Local Multiplayer Online on One Wii? Impossible

Not possible. Super Smash Bros. Brawl on Wii supports:

• Local Multiplayer: Up to 4 players on one Wii (GameCube controllers / Wii Remotes). Offline only.
• Online Multiplayer: 1 player per Wii via Nintendo Wi-Fi Connection (discontinued May 2014). No local co-op online.

Why?

• Netcode designed for 1v1/2v2/4-player FFA with each player on a separate console.
• Bandwidth/latency budget couldn't handle 2+ local + remote sync.
• Nintendo's philosophy: "One console, one player online."

Modern Alternatives

• Super Smash Bros. Ultimate (Switch): Local 2+ players can join online Battle Arenas / Quickplay together (shared screen or split).
• Slippi (Melee Netplay on PC): Dolphin emulator + Slippi rollback netcode. Supports local netplay (multiple controllers on one PC) vs. remote.
• Project+ / Project M (Brawl Mod on Wii/Wii U): Local only; no online.

For Brawl specifically: gather 4 friends on one couch. For online + local, upgrade to Ultimate on Switch.