This is not an issue of supply and demand, other than potentially more demand. If you raised the national minimum wage, then companies would almost immediately band together and charge more collectively, because they know that their products have all of the sudden become cheap relative to what the consumer can afford. Thus, there will be potential to raise prices to where they can still maximize profits without sacrificing demand, rahter than have high demand at too cheap of a price.
Wages will also trickle upward as prices do if they haven't already, and so yes that is where it can eventually lead to the Fed manipulating it with inflation, but it doesn't have to as long as higher prices can be maintained.
What does it matter if I now have to pay $1 instead of 0.50, if people are still able and willing to pay $3 instead of $2 for it?
That's where I think you're grossly oversimplifying some other prices needing to go down to compensate, when it's the higher wages that compensate for the higher prices, perhaps even across the board. There is no negative correlation unless the products are direct or indirect competitors, and as explained by Friedman, it is the priveledged who will be favored over the competitors in that situation.
Thus, you reduce competition, raise wages, raise prices and it matters little if you have to pay a little more to make it happen. If this happened in a free market, you'd call this savvy business, but in a manipulative market, it's just another sickening way the ones on top break the rules to stay there.