I think it kind of depends on how old your cousin is, and what kind of events will take place. Is this a party for a small child? Will there be cake and ice cream? Will the guests play games? Is everyone expected to bring a birthday present? If so, this is a birthday party at which a lunch or supper of barbecue will be served. You might call this a birthday party with barbecue.
Is this a gathering to celebrate the birthday of an adult? Will you encourage guests not to bring gifts? Will there be a cake but it is served only as dessert after the dinner, and there is no big fuss? In other words, is this an adult party? If so, I'd agree just call it a birthday barbecue.