I voted for Pig N' Out. It tells the customer what to expect and fits your business plan for now. It also has a double meaning and play on word that you were looking for.
Slogan: Pigging Out @ Pig N' Out.
Team: Pyle's BBQ; Yoder YS640; Good-One "Trail Boss" 60T; Homemade rig; 22" Weber Kettle w/rotisserie ring; Big A$$ Rotisserie; BGE-Small; Weber Kettle Gold, Weber OTS, New Braunfels offset smoker.
Saddles BBQ Bistro