I feel like Chekov might have something to say about this gun.

"And then this lady left a gun in my room and very pointedly told me I'm right where I need to be."

It was an odd shot too, like the camera focused on her putting the gun down. It felt like it was meant to, or is meant to in the future, come back, but maybe it's just a dangling thread.

Yeah I was wondering about that

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.

What if La'an leaving the gun in the room was a predestination paradox? Leaving it there means Khan rises to power, fathers more children than he would have otherwise, thus resulting in La'an being born in the first place.

…and when I was older I saw some security footage of a dude who looked a LOT like you…