I've never dared ride to a race--no guarantee what the old legs are going to be like afterwards. Get someone else to drive you!!!
I go to the gym occasionally on the bike, and you can slip leathers on over kit. If it's a biggish race there's usually bag storage, so you could use that, too. But let's face it: it's a nice day, you've had a good blast, there are miles and miles of virgin tarmac to be cut up . . . are you really going to get off that bike and shag yourself out running 13 miles when you could be blasting down the A5? !!!
Well done on the bike test, BTW. Fazer's a good first proper bike. Give it hell!