If you live in a tornado prone area, your BOB is in your safe room, and you hope you can reach it.
Working 40 hours away from the house, it becomes problematic. Many corporations prohibit firearms on the property. If you work retail at the Mall, no gun. Car parts, no gun. Etc.
Now you have the circumstance where your gun might be stored in the car, and you are not when an incident happens.
Take your BOB into work? Sorry, I've had nosy coworkers snoop thru it. The BOB or backpack carried by a male does not get the respect of a woman's purse, and is considered fair game by superiors for search. Despite the Fed.Gov furthering the BOB in urban use, the people are not embracing the concept. Keep a four inch fixed blade and a 30 foot piece of kernmantle in your pack and you are an active weirdo.
Security will show you the door on your way out and hand you your pack in the street.
One consideration about carrying a gun at all during an extreme weather event is the humans tend to band together a lot more harmoniously as the digging out and rescues are going on. There were almost no armed events during Joplin's EF5 recovery. It was us against the weather. But - it's not a urbanized area mindset, despite the categorization as a metro.
In your area, looting and riots after a weather event might be much more likely. Nonetheless, after Katrina, it took some time before it happened. Most people who could leave already had, it was push come to shove by those who stayed behind. And for some, that was a calculated risk so that the looting could take place.
Joplin post tornado had people driving into town to loot within hours. Not the neighbors, some came from another state.
One sad fact is that if someone attempts to shelter in a car from a direct engagement with a tornado, they are likely going to be killed. Your BOB isn't going to do much except bounce around in the car as it is picked up and slammed against the pavement becoming another object that will injure you.
In a severe event, the gun in the BOB at first is more a burden than help. It regains a place of usefulness when things quiet down and other humans start reacting selfishly. That may take hours or days. What will be more important is water, heavy gloves, extraction tools, and weather gear. Defensive weapons are further down the list.
You will more than likely not need one until the sun goes down the first night.