Beethoven's assistant was Ferdinand Ries, pronounced "Reese." The name was pronounced correctly in the film but incorrectly spelled "Reis" in the film credits. It was Ries himself who told the story of his incorrectly thinking the horn player came in early.
Unfortunately, the trumpets shown in the film are late 17th or early 18th century types after the Haas or Ehe family of Nuremberg builders, not classical era trumpets. The horns, however, appear to be accurate.