I think I was confused about the initial quetion. It's dawning on me that your question was whether a heat pump-only(no gas) solution was cheaper than an AC -only + gas heating combi-pack??
The answer to that would be "it depends". High end modulating heat pumps are pretty expensive, but code-min efficiency heat pumps with no frills can be pretty cheap. A so-so code min efficiency 3 ton heat pump system is about $1.5-2K at internet pricing. Gas versions are within a couple hundred dollars of that. But as with anything, higher efficiency costs a bit more. The heating BTU output of a gas version is considerably higher than the output of heat pump versions, but in SC the cooling loads are typically bigger than the heating loads (outside of higher elevation locations), and the heating output of most heat pumps at 25F is often higher than the rated cooling output. If sized for the cooling load, it will usually be oversized for the heating load anyway, just not 3-4x oversized like many of the gas burners would be.
But like any other heating/cooling problem, getting to the right solution starts with a room-by-room load calculation (Manual-J or similar). The chance to size it right only comes along when you are replacing the equipment. If the existing unit was cutting it but still cycling on/off on the coldest & hottest days of the year it's bigger than optimal. Right-sized units would run almost continuously for hours on the hottest or coldest days, but would be more comfortable than one that was cycling on for 15- 20 minutes, off for 10 minutes, except when recovering from a large thermostat shift, say, from "off" to "72F" the minute you get home from work. The oversized unit will reach the temperature more quickly, but then the humidity will start rising as it begins to cycle.
If history is any guide 3 tons of AC will be oversized for an average home, but it's hard to say how much. For some rules of thumb, a typical SC house with at least some insulation in the walls, and double-pane windows (or storm windows) that doesn't leak air like a sieve would have a heating load of something like 10-12 BTU/hr for every square foot of living space. Homes in SC that have at least R25 in the attic and not gazillion square feet of west facing window the peak cooling loads would come in at about 1 ton per 1000 square feet of conditioned space give or take. It gets complicated if the ductwork is in the attic above the insulation or in a ventilated crawlspace, especially if the ducts & air handler aren't well sealed, which can add a ton or more to the absolute cooling load. The old school rule of thumb was a ton per 500', which oversizes it every time, but takes a toll on comfort due to temperature under/overshoots and inadequate humidity control.
But rules of thumb like that are no substitute for load calculations. When replacing equipment it can be useful to work backward from energy use figures on specific periods, looking up the heating & cooling degree days for those periods, which is a good sanity check on any calculated number, or rule of thumb derived sizing.
OK, more than you asked. 'Nuff said.