Panzer III would overlap with Puma much. If anything, StuG III should be the "uber light vehicle".
A Panzer III Ausf. N (infantry support tank) would pretty much be a StuG E with a turret. It'd be more comparable to the AEC than the Puma.
I'd love for a Panzer III plopped in Ostheer T2, but requiring Phase Three tech to unlock. Ostheer currently doesn't get to play in the Light Tank / Heavy Armoured Car stage, they just jump from Scout Cars to Medium Tanks. It's the reason Mechanized Assault and Mobile Defense are so popular, a light vehicle that can participate.