The part as written is for a natural trumpet (clarino) in F. You could theoretically play it on any high trumpet.
There is some evidence that the part might have been intended to sound an octave lower then it is normally played. (See this discussion):
Thurston Dart postulated the part was intended for "Tromba, o vero corno da caccia" but this would put the part an octave lower than written. I believe there was a player, possibly even Reiche, who could have played it or had the required skill. Remember the portrait of Reiche holding a "Jaegertrompete?" It's impossible to say if such an instrument was common or not despite the famous painting as the only surviving such instrument was destroyed during WWII by allied bombing.
A feasible answer to the question "trumpet or horn?" might be that on occasion Bach allowed for a performance of the work incorporating a horn instead of a trumpet. The German name "Jaegertrompete" (hunter's trumpet) for a trumpet which was coiled, and looked like a horn, may have been mis-translated into Italian as "Corno da caccia" (hunter's horn), and thus the debate began.
If you do decide to play it on the B♭ piccolo, you don't need the fourth valve. And as for the choice between piston and rotary valves: that is entirely your personal choice. The part is hard enough, you should play it on the instrument you feel most comfortable on.