Is there any Hadith of Rasool Allah (SAW) in the Shia books of Hadith that describe the procedure for offering Salaat from Takbeer to Salam in a single narrative?
Yes. In the most authentic books of Hadeeth known as Al-Tahtheeb by Al-Shaikh Al-Toosi, volume 3, page 81, there is an authentic Hadeeth narrated by Hammaad ibn Easa. This Hadeeth has the procedure of Salaat from Takbeer to Salaam in a single narrative.
Wassalam.