How about an actual Solaris Game World RPG? Start a character in The Reaches fighting in exoskeletons and work your way up to BattleMech fame in the championship!
A story-driven format would be all right, although it's then just another Battletech game that's been done before, albeit in a new setting.
But you could have something of an open world where besides the matches, you have cloak and dagger amongst the stables, maybe such things as breaking into another stable's property to steal new tech they've been rumored to be working on (The Sasquatch, for example, made by the same company that would later create the Bandersnatch, I believe. From sourcebook The Reaches).
How about haggling with a tech or rival stable over used parts? Or even engineering your own upgrades? Although it would be very easy to break game balance allowing that.
Combat wouldn't have to limited to the arena matches either. Assassins, factional warfare among the Great House supporters, and even accidents could all happen in the streets.
A random mission from a database of missions could crop up specific to the area in which you're travelling, and you'd have all the immersion of the various areas of Solaris, flavored with the Great House each represents - or, again, out in the Reaches. I suppose the effect might be similar to the Jak and Daxter series combined with L.A. Noir after a fashion - you have the giant city with all its different sectors, and then the less settled and wild areas outside the protective walls - but much, much bigger.
I shudder to think, however, how long such a project would take. I envision an awfully large world.