I'm not sure where to post this but here goes: I have a 2010 Titan and it won't start in cold weather, anything below 20f usually. Got best battery available and new battery connectors yet nobody can tell me why this is happening. Alternator, battery, and starter tested fine so it's apparently a mystery. Also my belt squeals when it's cold out. 3rd new belt this year and 2nd new tensioner so yet another mystery.
Sounds like parasitic drain on the battery. If you can get your hands on a multimeter, you should be able to check the battery and fuses to rule out possible parasitic drain on your truck.