Barbecue is the correct American spelling. Trust me, I am from the American South and have consumed tons of it over my lifetime. Although there is a bit of disagreement about the etymology, the consensus seems to be that the word comes from the Spanish barbacoa. Horatia is right about BBQ being the culprit for barbeque. The Q stands for cue, as the syllable is pronounced, not as it is spelled. Also my Kindle (US) edition of Treasure Island spells the nickname Barbecue.
Even of it's an American spelling, why use it on UK products? I have uploaded a photo of the offending item to my profile and it clearly states that it is manufactured in the UK. And in the ingredients list they use British English spellings such as "flavouring".
I googled to check that (as I thought) Robert Louis Stevenson used the spelling 'Barbecue' for Long John Silver's nickname. The hits revealed an american 'crib' on the book using the spelling Barbeque - now that really is wrong, even if it is the US spelling they shouldn't alter names in literature.
On McDonald's barbecue sauce pots, they actually use the horrendously incorrect spelling "barbeque". This has been the case for several years, so not some glitch that got changed as soon as someone noticed.
This makes me despair for the future. How could anyone think that "que" would be pronounced "cue"? Think of all the millions of kids now thinking this is the correct spelling