I wouldn't worry about the shade of red; it's likely faded after 100 years or so.
Looks like an authentic cap for artillery or pioneer troops; the cockades are for a Prussian officer, but it is likely a senior NCO's cap, since it has a wool band. Officer caps have a black velvet band and senior NCO's couid wear officer cockades.
BobS
Actually the cockades are Württemberg.
Yes, I now see the traces of red on the cockade. Pretty well worn, though.
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