Many English-language jurisdictions combat alcoholism and promote the puritan work ethic by limiting bars to open only a certian number of hours per day, or to close at 1 or 2 A.M. So, right before closing, the bar tender habitually shouts "last call" meaning last reminder you can still order alcohol for about 15 more minutes.
And yes, Bar K. is a pun for bark, which is our generic onomatopoeia for a dog's yap. So, in anthropomorphising the dog, we are attributing to the animal a series of cultural characteristics that cannot exist in animals, so the creator appears to be fantasizing in a culturally recognizably erroneous fashion. Thus the humour of it.
For more information on this sort of explanation of humour, see The Origins of Laughter.