SS was a trim package in 1963; not a performance package. So an SS car will have bucket seats. If you got bucket seats with your two door hard top then yes it is a true SS.
As to motor being original you will need documentation to prove that.
In 1963 a four speed bought you a Borg-Warner T-10; which the 409 could make enough power to easily break, if the rear end held together long enough. It was the 409 breaking drive line parts at the hit of the throttle that motivated GM to produce a bigger version of the B-W T-10 using the same design, just super sized inside the aluminum case (wider gear face to transmit more power per tooth, bigger diameter bearings and shafts to hold the load).
Since it has bucket seats I would pull them to look underneath for a build sheet as it was easy to stuff a build sheet up in the springs as the worker carried the seat to the car to assemble it in the car as it rolled past on the assembly line. A build sheet was trash that was supposed to be thrown away, but a lot of workers were lazy and just stuffed them in the car out of sight to save a few steps.
I imagine the previous owner grew tired of replacing four speeds and rear ends and swapped in the TH350, which didn't come out until 1969, for the four speed to act as a shock absorber to make parts last longer (the torque converter absorbs drive line shocks that could shatter hardened steel parts).