Her hair will thicken in time. By the time they are 18 months old, you will for sure see their adult hair. It will probably thicken a bit more as time passes. Are you using a true rubber band? I have never had trouble with them coming out. Mine can't.
I use bows on alligator clips mostly. You do want to be careful with puppies. You don't want to leave them unattended with something like barrettes or clips in their hair that they can choke on. Just be patient. In time and with persistence, you will be able to put them in your little one's hair.