Sorry to be pedantic, but there weren't any Captains of Light Dragoons at Leipzig, I'm not even sure about the Corporal's Guard, and I'm sure that you don't want to make him a Corporal anyway.

How about he's a Captain of Volunteers (or more correctly, Yeomanry) and he's visiting relatives in Germany. He accompanies his German or Austrian or Swedish or Russian cousin (say) on campaign and gets involved in the Battle of Nations at Leipzig. He has his uniform with him because he knows his (cousin) is in the Regular Army and doesn't want to be shown up. (In fact half the reason for having a Yeomanry Commission was the chance to wear a flash uniform, usually designed by the unit themselves).

The particular Yeomanry he would belong to would depend where in the United Kingdom he lives (they were 'local' units), and the unit he joins on campaign would depend on what you want him to experience during the battle. I couldn't find an Order of Battle for the Allied foces at the battle on the web, so that might mean a visit to the library for you.