Having watched all of the matches and competitions between these two and the most recent (which I cannot reveal here) Ash is going to have a very rough time battling Paul in the Shinnoh League. It's not what pokemon Ash will use its incorporating them to bring down the toughest of Paul's pokemon.
One important aspect is the field:
If an Ice field, they can use the "bumber ball" technique Ash used on Pryce and Clark, or the melt the field.
For a water field, have Pikachu electrocute any pokemon in the water (duh)
And for Grass and Rock, Burn and melt the field.
Rock could have a debris shower if one of Ash's pokemon can learn the Rock Smash technique.
The second aspect is abilities of each of Paul's pokemon, Ash needs to get around each one that either raises power or creates a sheild. For the power users like Ursaring he'll need some crippling moves. An example is Grotle using Rock climb to comfuse him or if we are lucky use Swagger (this one's risky to use though) or better yet poison him. Now for the defensive moves. The best bet is to use projectile attacks instead of going head-to-head with melee attacks. What Paul will use in the league is unpredictable but I can see him using Torterra, Ursaring, Honchcrow, and Electabuzz.
If we are lucky, Paul's Electabuzz might evolve into a Electivire and disobey him.....though the likelihood of that happening is very slim.
