I'm with the majority, none of those really say catering bbq to me. If A n J are your initials or something, that's a good start, but I'd drop the House part, unless you are a restaurant. The n part is a bit weird as well. Something like AJ's BBQ would be simpler, easier to logo-ize (is that a real word) and easier for people to remember and suggest to others.
Neil -- "Easy Bake Master"
NB Bandera [gone to heaven...conveniently located in Long Island, NY]
Rebel 23, WSM A, WSM B, UDS, Easy Bake Oven
Pork Pistols BBQ
MOINK...It's what's for linner.