Shadowverse

As I understand it, this is a condition for receiving bonus cards, right?

What cards? If it's the ones from the DB/NE packs, then no, you don't need to complete anything, they are immediately available. The campaigns give you, if I'm not mistaken, only cards from the base set - two per class. And you can also farm some gold.

I'm not having any luck with this expansion.

I'm also not having luck with the cards in DB/NE.

I, on the contrary, had the worst luck with the Chronogenesis expansion. I played much more back then, so I opened more packs than DB/NE, but I got fewer legendaries.