Steve, I do know the FBI let Baker go. What I don't know is how much they had on him.
When you add up the clues, Baker is almost a perfect suspect. Who else but the superintendent at Plymouth (Baker) knew about "J./J. Faulkner"? Who else but Baker is linked to Ellerson via Armour Meat Packing and "Sha-Toe?" Parenthetically, who else but Ellerson knew the Lindberghs would be staying at Highfields on a Tuesday night for the first time ever? Who else but Baker left town shortly after the ransom exchange at St. Raymond's?
How much of this did the FBI know? Did they know Baker's wife was German, and could easily have written the ransom notes?
If Baker was innocent, then who used "J./J. Faulkner" as a alias? As I understand it, the FBI searched high and wide for a J.J. Faulkner, only to discover there was no such person. Ipso facto, "J./J. Faulkner" had to have been an alias, and who else but Baker knew those exact letters appeared on the Plymouth dumbwaiter?