The new 2014 Ram 1500 Bighorn pickup has a payload of 1800 lbs
Not sure where you're getting that number from? It appears that the Ram has improved it's payload across the board but there is only one with a payload that high, a base regular cab long box 4X2 with a V6. The Big Horn ranges from a max of 1632lbs to as low as 1077lbs.
See
here for all payloads by engine and body configuration.
The 1500 may be ok for the TT but it all depends on the configuration and trim level. Or just get the 2500 and have no worries.