Of all firearms, the handgun is the most difficult to master. If you doubt it, there are numerous stories where law enforcement officers and bad guys have fired many, many rounds at close range and no one got hit.
Since she has experience with the shotgun, it would be a fine choice. There are many choices of ammo beyond just bird shot or 00 buckshot. #4 buckshot may be a good choice. At the end of the day collateral damage is always the responsibility of the shooter, and it can incurred with just about any ammo. A good plan is imperative.
You may have already broached the subject with your daughter, but it is very important that she have the mindset to actually use a weapon against a threat, and be prepared for the fact that the end result could be someone's death.
I applaud your sentiment and if she is prepared, then by all means go ahead, but younger folks often haven't worked through these issues.