have 2009 R/T with remote start-
had the same issue with remote start turning on then off after 5 seconds. will not turn back on using the FOB buttons on repress. At the time, the battery was giving problems starting the vehicle in the mornings.
changed the battery (original 2009) to a new one. the problem went away, however it did it a few days later. i learned that problem will still be present if the battery voltage goes low. I never bothered charging the new battery. just swapped it and drove. i had to throw trickle charger on battery to peak it. remote has worked every time.