I think I have a good solution to the "Meat Monger" dilemma. How about "Salammier". Just like a sommelier is a connoisseur of wine, a salammier is a connoiseur of salamis and such.

Back in the 1930's people were often referred to by the tools of their trade. In engineering offices draughtsmen were called "pencils" or "T-squares"; people doing calculations were called "computers" or "calculators". This deli is trying to maintain a common man ambience (hence the original "monger" suggestion), so how about "slicer" or "knife"? The nice thing about these terms is that all PC issues are avoided.