Sami a really great video mate, and Imo you touched on some good points on the pros and cons of Sonic Adventure 1.
IMHO, I'd turn Big's 'stages' instead into a visual novel. One thing that's neat about SA1 is how each character's route, other than Sonic's, feels as an 'alternative look' at the game's plot and events. A side story that contemplates and expands on the game aside from what players see from Sonic's PoV. Think something akin to PS3's Folklore where the 2 characers's stages has them not directly meet but 'interact' in a way.
In the case of Big and why I'd turn it into a visual novel kind of thing, it'd be Big merely reflecting on the events of the game from his PoV while he searches for Froggy. Yeah it sounds pretty superfluous, but if you can't make fishing as a main gameplay system any better, but you need to keep Big, may as well make use of him somehow :V
His fishing stages could stay in as a sort of side mini game that you could play as any of the characters with.