Hello, each level is played singularly, so to complete a campaign you have to play 4 times. Each time you have 16 rounds.
I also thought first that you need to do all 4 objectives within 16 rounds. I managed to do it easily on first campaign, around round 10-11 with some luck on planets. Totalled 47 points (with all objective card points). Only afterwards I started to wonder about the points per level requirements and the bonus resources, but could not crack it solely from the rules.
This forum and publishers/designers answering questions here is just awesome. I tend to make rules mistakes in every game at first.