Prince Of Wales Marriage Medal 1863

The reason that it's not to be found in BHM is that it is a purely local medal. Local to Ross-on-Wye in Herefordshire where the hedgehog is the symbol of Ross-on-Wye. Ross was an important town in the ancient kingdom of Ergyng (AngloSaxon: Archenfield: Land of Hedgehogs).
The obverse of your medal is common, the official portrait by Spink probably. The reverse however is another matter. I collect and deal in such medals and once lived very close to Ross, but this is something I've not seen before. Possibly quite rare as the population of that town in the latter part of the 19th century was less than 4000 and it is likely that distribution was small and survival intat quite uncommon.

It appears to be by Ottley Birmingham with the signature below the POW's truncation.

The obverse die appears as 932 in Andrew Whittlestone and Michael Ewing's "Royal Commemorative Medals 1837 - 1977 Volume 1. Queen Victoria 1837 - 1901 excluding the Jubilees of 1887 & 1897." [Galata Print Ltd., 2008].

They record two pieces that use this die: 932A which is a generic one detailing the event in text, and 932B which is a local issue for Upton St Leonards.