There is another way for the silenced pistol to get back the Tower without being stolen. BUt, it gets more complicated.
Saul wants to take the Hummer. Burt convinces him they should take something smaller. I do not remember this being resolved. So, I will make the assumption they took a truck. When Saul and Burt exit the Furniture store, they call for backup. Riley, Michael, Angel respond. There is always the possibility of others. But, again it complicates things. They set up the .50cal in the hummer. When they find the mallers gone, the group fears that the mallers are going back to the tower. The team splits up. Saul and Burt investigate the mall. Angel, Michael and RIley come back to The Tower.
Victor says two trucks came back.
So, Michael Angel and Riley come back in a truck.
Saul and Burt now have the hummer. This is confirmed later when Saul asks Burt shouldn't we go back to the hummer after spending the night at the airport terminal.
When they swap truck for hummer, the silenced pistol may have ACCIDENTLTY fallen out of Burt's bag. Riley, Angel and Michael are still known to have access. We hear Datu ask for help bringing the weapons back up from the truck. Michael goes to help and Victor is asked to tell the guards to stand down and send a few down to the motor pool to help move the weapons. It could be taken now or it could have been taken by Datu. Although Victor is told to send a few more people to help, Pippen shows up and Victor tells people to get back to their post.
It is possible that it gets all the way to the armory.
It is possible that the shooter takes it out of the armory. Although it would have to be fast, the shooter might know the exact position, because they were their last night.
The above was based on the assumption that Burt and Saul leave with a truck and swap for the hummer.
But, there is still another truck. Like I said, more than just Riley, Michael and Angel went to the Mall. More Likely, it is the search team sent to look for lizzy before Saul and Burt head out.
Assumption Burt Left with the silenced Pistol
Assumption Burt/Saul leave in Truck and swap for hummer
Assumption:The pistol falls out of the bag.
Assumption: Datu, Riley Michael are the ones who move the weapons from the truck to Burt's Place.
This allows the Silenced Pistol to be shifted from Burt to the Shooter without the Shooter actually stealing the weapon from Burt, thereby placing bouth Saul and Burt in danger.