I'm going to be a bit of a naysayer here and lay out some reasons 'mech racing probably isn't much of a thing.
First, let's just start with the military aspect - how often do we have any type of races using military equipment? I'm sure there is the occasional non-combat private situation used as part of a training exercise or to let the troops burn off steam / boost morale, but it's not like we have Formula M1.
So in universe, where are 'mechs used for entertainment? We have Solaris, and the odd example of other arenas generally on mercenary heavy or pirate worlds, but considering the large populations of many planets in the IS, we don't see 'mech battle arenas as part of the on-world offerings in many places, which is part of why the Solaris matches are so heavily distributed across the IS. This probably means it's a safe guess to think 'mech based entertainment is actually pretty rare for whatever "reasons".
Also from the entertainment aspect, the general audience (and therefore money) obviously is there for the 'mech battles, but since we don't really have much on racing, it's another safe guess to say the people want to see them fight, not race.
Which brings us to the logistics of 'mech racing. One of the inherent weakness of all 'mechs is that they have a major weak spot in the rear, with various venting, access hatches, and other components minimizing the amount of armor that can be applied. So if we want weaponized racing, the obvious strategy is to be last, not first, so you can shoot everyone else in the back. Of course, everyone would aim for this strategy, and you now have a sluggish melee as everyone maneuvers to slow down and get behind the other racers (except for the odd occasion where a racer realizes everyone else is focused on that and takes off hoping to get away before anyone realizes what is happening).
So what could make 'mech racing interesting? Melee weapons/attacks only, so now you have a reason to get away from everyone else, and a reason to catch up to your opponents. This is potentially more expensive as you have your racing equipment literally ripping each other apart, but once we move into eras where 'mech production is ramped up, or the disarmament period of the RotS this would be a good place for industrialmechs or scrap battlemechs, and since we don't have to worry about stray weapons shots, this "safer" entertainment might actually be more prevalent then battle arenas in the right time perioda / planets.
But the audience really wants to see the guns... so how do we get around the rear armor issue? Can a race track / arena be designed in such a way to minimize line of sight such that a racer wants to get out ahead in order to prevent the opponents from having a shot? It would require lots of cover, few straight stretches, and you'd probably need to restrict jump jets. Limiting weapon choices to short or medium range weapons would also help, but then you lose the excitement of the big guns (except for the racer who somehow packs in an ac20...).
So, there are probably some races out there, somewhere, but if you're involving the expense of a 'mech, the smart money is in the arena where ultimately the glamour and fame of Solaris flashes C-bill signs in the dreams of the aspiring competitive 'mech jock.