Howard, I am not a criminal legal expert, but from my limited understanding, unless the police believe the father is an imminent danger or is going to run away, they lose nothing by "talking to him" to collect evidence. They do not need to arrest him to do this. The father may give statements and/or evidence without having been arrested and this will all be admissible if indeed a criminal conviction is sought afterwards.
Actually, in these types of cases, in the UK it is not recommended for police to arrest and take statements immediately because it gives a defence to the accused later down the line that they were not of sound mind when being interviewed.
In my opinion, absent an imminent danger, and assuming that this really is as simple as "SIDS" (and the act of accidental suffocation of a child is not a criminal act), IMHO the police acted too quickly and without sensitivity.