I think you should go with the balance bike, as you said she doesn't have a whole lot of leg strength, bike experience, etc. Especially if you are able to borrow one, or pick one up fairly cheaply. If she was really into riding already I might suggest just going straight for a "normal" bike.

And just to let you know, once DD is ready for a "big bike", there are different size bikes, even within the same tyre size. Ie, you can get a 16 inch bike (which is about 4 year old size) in a smaller or bigger size. The handlebars are different, and I think (but can't remember) the frames could be a little different. Once on a bike, if DD still finds it a little difficult to pedal, change the rear sprocket (the gear on the back wheel the chain goes on) to a slightly bigger one. Keep the original sprocket, you may use it later. I remember my Dad doing this on my first bike because where we lived was on a hill and I couldn't pedal up it.