Does the federal minimum wage impact the cost of living in any way?

Yes, and it's not hard to prove. Let's do a thought experiment, say we make min wage $50 an hour, and let's assume no jobs get automated or eliminated. Now for this example and to make the math easy, we'll say our cost of raw material is 0, and that the wage before the rise in min wage was 7.25 and it took one hour to produce our item and that we sold said time for $10 that is we marked it up my about 38%, so to keep the same profit our $10 item would now cost about $69. Basically you'd have steep inflation and the value of the dollar would go down and the "poor" would quickly be in the same boat as they were before only the number on the currency would be bigger.