Should the presidents have pokemon from their region? That would make logical sense, wouldn't it?
Yeah, I kinda have to agree. That and it'd be patriotic and all. (I could make a joke about birth certificates and US presidents and all, but I'm lazy.)
That being said, presidency's partly about image, right? So, it'd probably depend a bit on how you want the leaders to portray themselves. For example, if you have a president who wants to look powerful, they'll go for something a bit more sturdy and battle-ready than something that people would think of the regional mascot. (Like the difference between Tyranitar and Togepi for Johto's president. If they're tough war birds, definitely go for the Tyranitar.)
So, it's hard to say with the male presidents/ambassadors because you've got a variety of different kinds of leaders in those departments. You might have the war birds who want to prove that they're capable leaders in tough times, and you might have the diplomats who want to prove that they can make strong, intelligent decisions. With women, it also depends on how the world views women as a whole. For example, in the real world, when you've got women trying to get positions of power, they tend to try to make themselves look intelligent, diplomatic, but strong and firm because some voters don't expect women to be strong leaders. So, you might have something to represent that (like a Psychic-type people might think of as sturdy or reliable -- like Metagross or Claydol) if women are treated similarly in your version of the Pokémon world. If they're all equal, then eh. Same rules for men apply here, too.
Long story short, yeah, I can't really come up with specific ones, but it might help to think about how these figures are trying to portray themselves and come up with matching Poké-personas.
Also, definitely seconding the Orre bit.