Are you willing to double the cost of everything associated with the minimum wage positions to cover those increased labor costs? If not, then this is doomed to failure. If so, then the minimum wage worker is no further ahead than they are now - and thus it is still doomed to failure. A 6% profit margin simply cannot absorb a 50% increase in labor costs. This is basic math.
You assume there is a one to one correlation between labor costs and the cost of the product
There isn't
You assume there is no correlation between labor costs and the cost of good are services. There are.
So again - are you will to accept the increase costs of everything? If no, then this is doomed to fail. If yes, then the minimum wage worker will be no further ahead than before the increase because there will be an increase to goods and services. The only thing that increasing minimum wage does is increase inflation. It does nothing for the minimum wage worker (if it did - this wouldn't be the 9th time that minimum wage was increased in my short life time).
Now you are getting more realistic
Factors like rent, cost of supplies, taxes, advertising, insurance also contribute to the cost of your cheeseburger.
Raising labor costs will affect your cheeseburger just like everything else in business. Labor costs also reflect more than the cost of a minimum wage worker
I do not expect the person frying my burger to work for slave wages just so I can save a few pennies