The real answer is, not without a complete redesign from the ground up. The Final Fight sequels used a different engine because the original could only display a handful of characters onscreen at once. Just as with the SFII, King of Dragons, and Knights of the Round ports, Final Fight 2 & 3 were also massively letterboxed. SNES Final Fight was fine for the time, but in comparison to the arcade it was a very different game. Like many of the earliest games on the system, FF is so fondly remembered because of its release date, color palette, and the high quality sampling possible with the SNES's audio co-processor. With emulation and the last generation's Capcom Classics Collection, there is no reason to ever play SNES Final Fight again; other than to see how bad it actually was.

The missing player character problem was a memory size issue which could have been fixed with a larger, more expensive cartridge. However, the same cannot be said for the missing stage. All of the moving parts of the background that interact with the player and enemy sprites would not have been possible in the original engine without massive amounts of slowdown and flicker; the latter issue being something that emulation has smoothed over for later generations of players. The same goes for the two-player mode. An "arcade perfect" version of FF on the SNES would have needed smaller sprites, letterboxing, and/or massive amounts of flicker to even run properly.

While the SNES receives well-deserved praise for its library, it was far too slow to handle a "real" version of Final Fight until late in its lifetime. This is why the Sega/Mega CD version was the better conversion, regardless of its color dithering. Other than the soundtrack, it could even have been released on a cartridge. Just look at The Punisher, or any of the Bare Knuckle/Streets of Rage games for more evidence.