This may or may not be a bug, but this makes the quest battle impossible. In the very first battle Stunky runs out of Sucker Punch and switches out, then returns after Frillish is defeated and uses Struggle. The guide suggests that Stunky should be debuffed, but it shakes off the debuff with its switch and then beats what's left of my pokemon with one Struggle. If this is an intentional change to the quest, it seems too hard for the first battle.

Hey, its not bugged - the AI was changed after the release of this quest which makes it switch out the Stunky.
The quest can still be done with some simple adjustments to the strategy you found on a guide. Just keep using Growl on the Frillish as well until it struggles, then finish it off with Peck. Then use Growl on the Stunky again until you can finish it off with one Ember.

Thanks, but it's still not possible to complete the first battle because Bergmite has rocky helmet and Growl runs out of PP before I can force it to break its Sturdy with Struggle.

Its still possible if done correct. I have done the quest just recently on 2 different accounts.

Is a speed boost Torchic necessary then? Because Stunky outspeeds it and causes too much damage to survive the rest of the pokes.

No, I did it without Speed Boost both times. You are either doing something wrong in terms of the strategy or your IVs may not be high enough, try to aim for high IVs.