Alarm might scare off the BG, but it's real value is in warning you that the BG is there so you can take action. If you are unarmed there is little action you can take other than dialing 911 and hoping they arrive in time. Also, you have no ability to confirm a suspected false alarm other than walking around the house unarmed. NOTE: Keep a cell phone near the bed, the BG might cut the power and phone lines... I would at least consider it were I a BG. An old phone that's no longer active will still dial 911, so you can put it by the bed and not worry about leaving your active phone in your pants or whatever.
If you have other firearms available, I'd get some kind of alarm. All the guns in the world are useless if you wake up with the BG standing over your bed. Doesn't have to be expensive, those magnetic kind that alert to a door or window being opened can be placed on rarely used entrances and windows and simply left on. Change the batteries when you cance your smoke detector batteries. Put them on your bedroom windowns and the bedroom door, leave the window ones on and turn on the door one when you go to bed... only one thing to remember and it's better being there than nowhere.
Since you already have pets, have you consided a dog? It doesn't need to be a huge maneater... tho I admit mine is
... many small dog breeds are very smart, loyal and courageous. They can be trained, if handled properly, only to bark at intruders actually entering or approaching the house instead of all passers-by... and once trained you don't have to remember to turn them on. Even a small dog can be a distraction for the BG during a break in or home invasion.
Now to the real meat of the issue: It is NOT difficult to argue with her logic as it is flawed. If she is unwilling to make the effort to even turn on an alarm, you have a problem no matter what you decide to buy. Self defense and protection requires effort of some kind on the part of the protected. Situational awareness requires effort. Turning on the alarm and testing it periodically requires effort. You do not merely point a shotgun. It is a tool that requires proficiency if not a mastery, and mastery is nice in a firefight. Firearms proficiency requires effort.
The reason for using shotguns for HD is that you can tailor the load to your situation NOT because it does not require aiming or skill. #4 shot is best overall for HD as it peneterates enough to kill the BG without having a miss go thru the 3 houses next door. It usually penetrates the equivelent of 2 walls; ie: the wall of the room it's fired in, and the wall of the next, usually entering the third room with greatly reduced energy and greatly reduced injury potential. Birdshot does not penertrate enough to reliably disable an attacker, 00 buckshot penetrates as much as many handgun rounds. Rifles are right out, many will go thru 3-4 houses worth of walls before stopping. Slugs will go thru fewer walls than a rifle round, but are still out for most HD scenarios. Keep a few in a side saddle for special situations, such as hostage or the rare long-range shot. These are values based on 12g shells. 20g is probably similar.
One option might be .410 bore. Mossberg makes at least one of their Persuader models in .410 bore:
The recoil would be easier to handle for a novice and penetration might be less while still being sufficient to disable the target. Winchester makes a .410 load with 3 000 pellets, and I'm sure there are others suitable for HD.
However, none of this matters if she isn't willing to turn on the alarm, practice with the shotgun, and actually give some thought to her own protection. Deciding what to do on the fly is a lot harder than having thought about it ahead of time... and having given it some thing it will probably be less of a shock when something happens.