I would suggest looking at the history of minimum wage increases. As far as I'm aware, there isn't a correlation between minimum wage hikes and a lagging inflationary period. Empiricism trumps theorizing about what "should" happen. Or, as stated by Hume, "You can't get an ought from an is."
In a word, yes. Inflation is a complex system effect that has a lot to do with many factors including money supply and confidence in the government, federal reserve, markets, etc. A minimum wage increase doesn't effect the money supply, because it's businesses who are paying, not the fed. What it would do is to cut into corporate profits, whose wisdom one can certainly argue about. I think the data indicate that it would help, not harm, the broader economy.