I know it sounds crazy that Dad's story could be true... but we've seen crazy things happen. I hope the police are not falling into the confirmatory bias trap here. They need to keep an open mind even if they are personally convinced someone is guilty. It's imperitive if our justice system is to work.
If the Dad's story is actually true then they need to find that baby! They change so fast and in a couple of months he will look totally different. He needs to be found quickly regardless of the circumstances.
They need to run parallel investigations here. One assuming the story is true, looking for an abducted baby. Another if they believe there never was an abduction. This doesn't strike me as the type where a parent got rid of a child... but I admit the abduction is also an unusual circumstance... so I just hope they have rock solid evidence of whatever they decide happened here.
In any event, LE needs to look for the baby.