A theme park gift under $10? Theme Park Insider: 2016 Year in Review
I think either of those days, Jan 3rd or 4th, will "probably" be about the same.
Because it's a Wed. & Thurs., you'll avoid the weekend rush (and the New Years Day crowd)
While Jan. is typically slower than the rest of the year, because you'll be going that first week, you may still run into the last of the holiday crowds (Some kids will still be out of school & many people will extend their vacations through that week)
I hope you & your daughter have a wonderful time! You'll still get to see the Christmas overlay (it runs till Jan.6th)
Also, because she's a fan of the princess characters, make sure you do the character meet & greet in front of "It's a Small World" (There is also wheelchair access here, just ask a cast member for assistance)
Walt Disney World
Tokyo Disney Resort