But apparently that is exactly where it is.
It was given to Tsar Ferdinand I (Coburg) of Bulgaria and was in the 'Old Sofia' Museum until recently which contained many personal possessions of the Bulgarian Royal Family. The museum was being moved and I don't know where the carriage or the museum are now.
I haven't discovered who gave the carriage to Ferdinand but he is also said to have given his wife, Princess Marie Louise of Parma sapphire and diamond earrings which once belonged to Marie Antoinette. And it doesn't end there.
Ferdinand also inherited the 18th century Antol Manor in Slovakia which is now a museum. The manor features a gold salon which is referred to as the Salon of Marie Antoinette as it contains the furniture Maria Theresa gave her youngest daughter as a wedding present.
Hopefully I will discover by what means that arrived in Slovakia soon.
Ferdinand seems to have made something of a hobby of collecting bits and pieces associated with the French Court.
He bought the pediment of the right-hand wing of the Chateau of Saint Cloud when it was being dismantled for his house in Euxinograd.
(Apparently the pediment from the left-hand wing is on the Chateau de Jeurre near Etampes, the main portcullis is at the Chateau de la Punta in Ajaccio and the bas-reliefs from the staircase are in the Chateau de Laeken in Belgium.)However here are some more photos of the carriage from the Alexander Palace Time Machine site.