I'd say it was okay for you to claim back your third, too as it's just mileage you'd be claiming anyway and as long as you're not required to fly - just doing it as an alternative method of getting there if it's possible on the day - then it's not a problem.
JAFO, this is my understanding also.
In the same way that you could reclaim a rail ticket, a 1st class flight etc.
Particularly as you're "just a regular employee".