Greetings Jim and very good question. I have asked the very same one many times to different instructors. I wish there was a definite answer but it really depends on their personal opinion and wishes. So I would recommend going with your instructors wishes.
Some would agree with training with the configuration you will use and others would like to begin with the basic configuration that most all starting OW divers in the recreational world are employing. As far as your daughter is concerned, both of you need to be totally comfortable with both of your gear configurations. By that I mean each of you need to know where everything is located and familiar with alternate location and deployment procedures.
Now with that being said, IMO ,"in my opinion" I understand your desire to go with the long hose, bungee alternate under the chin. I also echo your desires and am switching my standard rec-rig to that configuration. This is my single tank regulator set up for my personal diving which is limited due to I dive doubles primarily for my personal pleasure dives. While assisting training as a DM I have been asked to maintain a standard regulator configuration to make it easier for students and be more uniformed to shop rental regs. It just makes sense for a lot of reasons.
Once your OW training is complete I would switch if you both so desire. At this point I would train till both of you could perform air share drills effortlessly. Bear in mind that if you dive with anyone new you will need to explain to them what air sharing procedures are. Not a big deal, be prepared to answer questions.
My recommendations would be to gain experience and familiarity with your gear as well as diving in general and then sign up for AOW. I would train in the long hose, bungee under chin configuration. Practice your 18 basic skills till they are habit and do not take them for granted. Mask skills, free flow, etc. can and will save you in a emergency situation. Enjoy the journey diving is a incredible experience and to share it with your daughter is twice as awesome! Memories are priceless and being in your shoes right now with my own daughter, see if a DM will take some photo's of your training. It makes it even cooler.
Have a blast and love every minute, keep us posted.
CamG Keep diving....keep training....keep learning!