A price increase of 4% on an already-premium-priced burger isn't going to drive away all your customers. Nobody stops going to a burger place because a burger went from $13 to $13.52.

I don't really buy the 4% number in this context. I could believe it for a large enterprise like McDonalds that can absorb the cost increases by virtue of a huge, vertically integrated chain of production, but smaller businesses usually have far more middlemen in their supply chains, each of which adds overhead, along with the actual restaurant staff. Plus, it's not just the burger costing more - it's a ton of items, and in the end it may not be financially prudent to have the luxury burger anymore.

(Of course, this is me pretending that people who earn minimum wage are often financially prudent; in reality they'll just buy the burgers on a credit card and worry about it later)

If your supposition is correct, why should we, as a matter of public policy, protect the businesses that aren't efficient enough to pay employees adequately?

It's often argued that nobody can afford to, but here you seem to be assuming that it is possible, but it's important to preserve the companies that are uncompetitive.