The Blanche captured the Pique in 1795 and this famous engagement is well documented. The ballad would have been written shortly after. BVB appeared about 20 years later and not so widely printed as Blanche, probably first printed by Birt but continued upto Such's time about the middle of the century, whereas Blanche was much more widely printed, one of the first possibly being Morren of Edinburgh, or Howard and Evans of 42 Long Lane, London.

Unfortunately the Blanche was recorded from oral tradition in the early years of the 20th century but the collector neglected to take down the tune.

FWIW the Blanche fits perfectly to the tune sung to a similar ballad to BVB, 'Ratcliffe Highway'/'Rolling down Wapping' Roud 598 which has a superb variant in 'Royal Earsdon Calling on Song' as sung by The Watersons, one of my favourite tunes.

It might be worth looking through the many printings of the Blanche to see if another tune is designated.