First, do you keep the carbine bedside so that in the case someone is in the house at night, you can get to it ASAP? How about quickly accessible during day hours? Not that many don't, but a lot of people I see talking about using their rifle for home defense also keep it locked in a safe. My first thought would be, if the rifle isn't reachable in a HD situation, don't worry about buying an optic that is or isn't HD friendly.
Second, how do you plan to use this/any optic or sighting device should a HD situation come up? What if it's day time? What if it's night time? If it's night time, even if you can see a red dot can you identify what you're about to shoot?
There are a lot of people who have put time, thought, and training in to how to use a carbine for HD. So many other things than the specific optic used are more critical to survival that the question is a bit unimportant. I'm not one who has spent much time or energy on the subject but just enough to understand the basic bullet points. What I've found is that we worry way too much on specifics in the gear rather than the skills to not die. 5x is too much, but I'd rather have a carbine with a 5x scope on it loaded and in my hands than the perfect optic mounted on a rifle that's locked up in a safe with ammo in a different room.
If you're really interested in using the rifle for HD, take a HD class from a reputable instructor. You'll learn more in a weekend of class about what will keep you alive than you'll learn from a year of debating gear specs online.