Sometimes, it seems like they forget about the spore drive. They could have leapt to the other side of the galaxy the instant the away team beamed back from the Archive. The Breen might have the ability to destroy the Archive, but with Discovery gone, it wouldn't gain them anything and how ever long it took them would be time lost for chasing Discovery. When the Breen catch up, Discovery could just jump to the other other side of the galaxy.
Also, why didn't Discovery prepare a fake clue? With 31st century replication, it couldn't have taken more than seconds to prepare a reasonable facsimile of the original. Moll did no more than a visual inspection of the artifact to affirm its authenticity and she had never seen the completed object, only some of the pieces. The real artifact is a small obelisk with a button on top. So, make a small obelisk with a button on top that projects coordinates to a random star system. There would be no way for the Breen to discover the deception except to go there and find nothing.
The Breen might have the ability to destroy the Archive, but with Discovery gone, it wouldn’t gain them anything and how ever long it took them would be time lost for chasing Discovery.
That's a hell of a gamble if your objective is to protect civilians (and likely a gamble they would lose, considering the Breen's actions).
why didn’t Discovery prepare a fake clue?
I think the answer is the same - it's not something you want to screw around with if you're trying to save lives.
They faked the ship's destruction, letting the Primarch think he could pursue the objective at his leisure. He could take his time being petty and destroying the Archive. Discovery visibly escaping would put him under time pressure. If he delays to indulge his pettiness, Discovery could get the technology. They'd be using themselves as bait to lure the Breen away from the civilians.
Also, the Primarch only made his threat against the Archive later. If they'd jumped immediately after the away team was aboard, it would have been before the threat was made. It also would have been before the Breen's weapon demonstration, so the Archive would have been 100% intact when they left. At that moment, they had every reason to believe that the Archive was only in danger due to their presence. The logical, civilian-saving response would be to remove their presence.