Yes and there will be the usual political consequences associated with inflation; but this type of inflation is caused by a tax and cannot be combated by raising interest rates.
Why wouldn't a rate hike make a difference? It will lower demand and therefore prices, no? I mean, this isn't really something that we should celebrate or want, since it essentially just means discouraging people from buying shoes because they can't afford it, but it does bring the prices down (or at least slow the rate of shoe price increase).
This is blatantly false. You just have to look at Jerome Powell's reasoning in 2018-2019 and just last month!
It most certainly can, though you would have to push interest rates much higher than normal to kill demand enough to have an effect.