Looking at a few studies online, it seems as though there is no long term correlation between increasing the minimum wage and inflation.
Just so it's clear I think if there is a legal solution to this it's on taxing the high end of incomes, inheritances, and capital gains, not the low end.
If there were a long term inflationary effect, then industries particularly sensitive to minimum wage costs should have significantly higher prices in Australia compared to the US. That doesn’t appear to be the case; perhaps they absorb the additional costs in the profit margin. It can’t have no effect, but it’s clearly not as simple as you were implying.
There are offshore clothing manufacturers that arrange the clothes on racks and pack them in the containers.
Once the package reach the destination they do directly on display. This "saves" the company from having to pay someone higher wages for arranging clothing displays.
The companies will always chose lower costs wherever they can get away with.