Why did the older Harry not take the younger Harry with him?
Author's response
Several reasons:
Neither Harry nor Christine know how to take care of a child.
Harry has his mission, and Christine has to follow him, so one can't stay home and watch him while the other works.
They obviously can't take young Harry to work with them.
Since all of the people Harry used to know are either toddlers or haven't met him yet, his only options are Hawthorne and POSSIBLY HMO Hexlie, although she hasn't been developed enough as a character to be a viable option. I imagine it's also not good form to ask your boss to babysit for you.
Christine, with her abrasive personality, probably doesn't have a plethora of friends to ask either. She could possibly ask her parents, but her suddenly showing up with a three-year-old child might raise some awkward questions, even if they know it's not hers.