The way we have things rn is actually how it is ordered in the novelization. This works better because:

It makes no sense that they could identify the necklace practically the second Kylo grabs it. At least with the gap it’s slightly more believable.
Poe says that there are always random First Order patrols in crowds like on Pasaana. There are mostly Aki-Aki there, so when this patrolling trooper sees a strange group of people running, it makes sense that he would ask them to stop and show some identification.