I came up with a scenario like that a while back. The problem comes when you compare Matt 2:3-6 with Luke 2:22-38. The events in the Matthew passage would have to have taken place long after those in the Luke passage. By that time, after all that went on in Luke, all the Jewish leaders and "all Jerusalem" should already have known that the Messiah had been born a long time ago. Any troubling of Jerusalem should have been over and done with when Jesus was presented in the temple shortly after His birth. In Matthew 2:5, the chief priests and scribes would not have had to refer to a prophecy to discern the location of Jesus birth; rather, they could have told Herod "Oh, yeah, our Messiah -- he was dedicated in the temple last year." The whole tenor of the Matthew passage indicates the Jewish leaders didn't have a clue as to what went on in Luke.
There's also the problem of why Mary and Joseph would make the 65-mile trek back to Bethlehem from Nazareth when Jesus was a few months old. In my scenario, I have them going to Jerusalem for the Passover (Luke 2:4) around a year later, and going the extra 5 miles to Bethlehem to visit the people who had taken them in during the 40-day purification period after Jesus' birth. (Hopefully they hadn't had to stay in a filthy cave all that time, and Matthew notes they were in a house, not an inn, when the magi found them.)
One thing in favor of your/my explanation is Herod's ordering the killing of children as old as two years. The dumbest of Herod's grunts should have been able to tell the difference between a walking, talking two-year-old and a newborn. He was more likely looking for a one-year-old and "bracketed his shots" with respect to time to make sure he got him.
Another possible wrinkle is Archelaus (Matt 2:22). God warned Joseph not to go to Judea because of him, yet, per the NIV Study Bible, he reigned for ten years, from 4 B.C. to 6 A.D. That doesn't leave room for many yearly visits between Luke 2:41 and 2:42. Perhaps in later years Archelaus became too occupied with other evil deeds to worry about the threat of the Jewish Messiah?
Ra McLaughlin is Vice President of Finance and Administration at Third Millennium Ministries.