Intermittent Problem on Start Up of Passat

Hi. I have a 2003 VW Passat. The car has an intermittent problem where it will not start. The battery is 11 months old and the dash lights up so there’s juice there. Twice it has died, twice I have had it towed to two different repair shops (the second being a VW shop), and both times it starts on its own at the shop. In early August I got caught in two wicked rain storms but the car has had the opportunity to dry out. Both repair shops gave it a look over and said all connections are fine. The second repair shop put it on the computer and nothing came up. Last oil change was early August. People have said it may be the immobilizer (although the light doesn’t light up on the dash), the ignition coil (wouldn’t the mechanic have checked that?), the starting relay (again, is that something a mechanic would hve automatically checked) or the crank shaft position sensor (my sister spoke to her mechanic and that was his suggestion). I’m out $400 and no closer to an answer. In January I had the engine replaced with a 2004 VW Passat engine. Could the problem be related to this? I’m at my wits end as I can’t trust the car and I can’t let my kid drive it.

Gasoline or diesel engine?