Honestly, I think there's 2 ways to think through this.
Way 1:
Magically the minimum wage is increased, and everything about the legislators stays the same.
This would increase inflation, as what's causing inflation is the lack of legislation and enforcement. Thus allowing companies to raise prices and profits unchecked.
Way 2:
The legislators change in such a way that it's logical and possible to raise the minimum wage. Also logically other legislation would be passed to reduce the unchecked greed.
This would not increase inflation on it's own, and likely would keep it to a healthy minimum.
I think most who complain about the minimum wage talks can only imagine the first way.