We've been blessedly free from being crime victims, but I grew up looking over my shoulder because of the neighborhoods we lived in. My daughters are close to yours in age and we've had the talks about "good strangers" and "bad strangers" early and often. I don't want them to have a complex like the one my parents gave me that everyone you don't know is to be feared, but I want them guarded and teach them to listen to their instincts.
With that background, when the question about my gun inevitably came up, I was able to say "To protect us from bad strangers" without missing a beat. I don't know what conversations you've had with your daughter, but I would suggest building on them. I understand with you having been the victim of a home invasion, you don't want her to regress by impressing on her that the world is to be feared. It's easier said than done, but you carrying and the home invasion can make her stronger rather than more fearful.
What other gun experience does she have? I was taking mine with me to the Tanner Gun Show since they were still strapped to my back. It's never too early to normalize and teach gun safety, IMNSHO. Maybe giving her positive gun experience will desensitize her to their relation to violence but allow you to preserve their defense aspect? I.e. daddy = protector, guns = daddy activity, guns = fun with daddy the protector.
Spend some time at http://corneredcat.com/. It's largely a "by a woman, for women" site, but it is good at embracing the "mama bear" side of self-defense that, IMO, helps when thinking about raising kids, particularly girls, who don't need to feel like they're in SEAL training.
If I ever start thinking about not carrying around my kids, I reread this story: http://corneredcat.com/Why/walmart.aspx
DEFINITELY read this section specifically about kids before you hit the range. Should give you some ideas for conversations to have with her BEFORE you go. http://corneredcat.com/TOC.aspx#Kids