I just re read your post and finally understand what you were saying. Let’s not pretend I’m an expert in reading comprehension. I’m clearly an idiot.
Ah, the Noodle Incident? 
I’ve seen “barbeque” on signage in Texas for the last 40 years. BBQ is from barbeque, not the other way around.
That’s okay. I have the same brain farts here all the time. To clarify to others, as it may be confusing:
On the official packaging of the products in question, there are two ways the word is spelled: “Barbecue” or “BBQ,” including Cole’s own brand.
The only “barbeque” I could find on the website is in some of the text descriptions of the product, written by I assume a Coles employee or contractor.
ETA: Actually, now that I look at it again, I should correct myself: There is one “barbeque” on the packaging: that is on the rice crisps. The sauces , seasonings, and other snacks otherwise all have “barbecue” or “BBQ.”
That doesn’t necessarily prove anything definitively. The term “BBQ” dates back almost 60 years to at least 1956. Unfortunately, “barbeque” doesn’t have a listing in that etymological dictionary. Maybe samclem has some resources that can help.
ETA: :smack: Ah, I see in his last post that he does have cites from 1850-1900, and it’s in the 1950s where the “barbeque” spelling picked up steam because of “BBQ.” So it seems clear to me that it’s “BBQ” that influenced the increasing popularity of the spelling “barbeque.”
Using Google Ngram Viewer, I see that “barbeque” is used frequently throughout the twentieth century, and is recorded as early as 1872.
“Barbecue” seems a little more common, and goes back to at least 1833.
“Bar-B-Q” doesn’t start until 1920, and “BBQ” about 1940. (A lot of uses of “BBQ”, especially early ones, appear to have nothing to do with a style of cooking.)
Don’t you mean “who’s”?
My experience seems to be that “barbecue” more often refers to outdoor cooking on a grill while “barbeque” refers to a flavoring that in some magical way is supposed to remind one of barbecue. E.g., barbeque potato chips. Like the way they’re “Froot Loops” and not “Fruit Loops”.
Just so long as they use the Cueens English.
I would need some evidence beyond the severely misspelled crossword puzzle to make that determination. 
Yes, those who are saying “barbeque” is the English spelling are wrong. It’s “barbecue” over here too, except in some trade names etc.