Here is my story idea:
Big gets framed for stealing some high-value Babylonian artifact. Now he's got the Rogues gunning for him because they think he's muscling in on their territory.
Big draws a lot of attention to himself trying to run away. Of course, the moment Rouge finds out that there is another piece of Babylonian treasure she wants in. Rouge knows Big is way too simple to pull off a real heist. She realizes whoever framed him has the actual treasure. So she decides to "help" him, keeping him safe from the Rogues while secretly using him as bait to flush out the real culprit so she can swipe the artifact for herself.
It gives Big something to do, it lets Rouge be the manipulative treasure hunter she is, and it gives the Rogues a plot that isn't just "lose a race to Sonic."