Here’s an alternate take. Keeping in mind that La’an is a security officer trained in investigative methods.

La’an could hear the voices and footfalls of a security response team. She knew they would be there in moments.

The investigators would be looking for the gun that shot Kirk and 3 guards.

With that weapon recovered, the search would focus on the missing shooter who they would have on security cameras in the foyer.

Without the weapon, there would be another level of scrutiny, and such scrutiny might draw attention to the Noonien-Singh project that might be enough in itself to alter history.

By leaving the weapon in plain sight, with her own markered DNA on it as well as Sera’s likely-tweaked Romulan DNA, she’s left enough for the investigators to look beyond the facility for their escaped killer without actually revealing how and where she and Sera exited.