Any predictions on what type of person they suspect?
It sounds like they have someone very young in mind, maybe someone who was college/seminary-aged at the time of the crime.
If it were to turn out to be a pastor's son, would that count as a betrayal of the community? It seems like they were hinting at a connection to the religious community.
This is a wild theory, but what if the "two weeks" was mentioned because they can't get the guy without the cooperation of one of his family members, and that family member has agreed to co-operate only if he can have two weeks to get his son (or younger brother, or nephew, or whatever) to confess.
I admit that it's a silly idea. I don't believe it myself; I'm just thinking out loud.
It’s possible that LE might talk to the suspect and anticipates that they may get a confession? They may think two weeks is a reasonable estimate of how long the process will take. Or yes, possible that someone in a position of influence over the suspect is hoping to assist LE and needs some time.