If the cash register can manage to figure out granularity in the local taxes, so can the price tag printer. Updating tags every couple years after an election might be a pain, but it isn't exactly a herculean task - they seem to manage it in places where tax-included pricing is required.
The problem is, as you say, no one wants to be the hero: No business wants to appear more expensive than competitors, and no legislator wants to be the one to make cost of living appear even higher.