The Latin legend bronze coinage is most likely to be ascribed to the city of Edessa. Fergus Millar " The Roman Near East " page 152 describes the "liberation of Antonina Edessa the renowed Colonia Metropolis Aurelia Alexandria quoting a Syriac document. As one can see the Latin issues of Caracalla has all those titles present except for the name Edessa. As far as I know the title Alexandria is not associated with Carrhae. The attribution to the city of Carrhae in the BMC is based on a coin that has the further CA within the legend (page xo in the introduction) I have never seen one nor has anyone I know who collects Mesopotania has ever seen one as well.
Interesting. This seems to be in accordance with the Dandrow article I linked: https://www.academia.edu/31013674/T...in_Coins_of_Caracalla_from_Edessa_in_Osrhoene Apparently these coins being from Carrhae comes from a scholar (?) named Eckhel who based this on a single specimen in Vienna (back in 1828). Dandrow cites recent Syriac documents, coin inscriptions, etc. giving what sounds to me like a pretty well-thought-out reason for Edessa as being their town of origin.
I used to collect the coins from the mint of Carrhae and discovered this problem when I acquired my copy of Millar probably in 1998. In 2003 I traded my collection for other Greek and Roman coins to a collector and a good friend who collects the coins of Mesopotamia. I am glad that somebody did a really good job of putting this question to rest.