The research quoted in the comments indicates that this is indeed a bad idea, and wouldn't accomplish the intended result, but if it were decided that the CO2 should be vaporized, it's highly unlikely that active methods would be the most economical.
By "active methods" I mean that we have to generate the heat to do the vaporizing, by means such as combustion, nuclear reactions, etc. A "passive" method would be one that redirects or concentrates naturally available heat, such as solar energy, geothermal energy, etc.
In my answer to this question I summarize and provide links to sources regarding the concept of a "pole-sitter" orbit, a highly non-Keplerian orbit. Such a pole-sitter can use a solar sail to remain in view of a planet's polar region, as shown in Fig. 1 below. I've omitted the gravitational force due to the Sun and the centrifugal force from the heliocentric orbit because unless the pole-sitter is a long distance from the planet, those are relatively small forces. Also, the discussion is about the CO2 reservoir at the south pole; assume in my figures that north is down.
Electric propulsion can maintain a pole-sitter orbit too, but in this case we'd use the reflected sunlight from a solar sail, and there'd be no propellant required, a significant advantage if the project takes centuries.
In this case, the sail can be designed to concentrate its reflected solar energy onto Mars's south pole, increasing the average temperature there and eventually vaporizing the CO2. "Eventually" probably means decades or centuries; it depends on the sail size. Such a sail wouldn't need much of a spacecraft (in terms of mass) to control it, so the total areal density of the system would approach the limiting areal density of the sail and its support structure.
This low density is important for two reasons. One: this sail will be big. When you talk about vaporizing 12,000 cubic km of CO2, any method you use will involve something on a grand scale. The amount of sunlight you'd need to redirect to the pole will be very large, and that requires a large reflecting area. Lower density means less mass required to build the system. Two: the lower the areal density of the system, the higher in angular elevation above the pole's horizon a pole-sitter can get, so the energy distribution is better. In Fig. 2 below, this means the angle between the green "viewing angle" arrow and the planet's rotation axis is smaller. It might wind up that because of the increase in structural mass fraction as the sail goes beyond a certain size, a constellation of smaller sails might be better.
There is a problem the designers would face. The Sun-planet-pole-sitter geometry doesn't depend on the planet's obliquity, the tilt of its equatorial plane with respect to its orbit plane, so the pole-sitter is always a bit farther from the Sun than the planet. This is required to have all the various force vectors (gravity, centrifugal force, sail force, etc.) add up properly. That geometry rotates with Mars's revolution around the Sun, to keep the pole-sitter on the anti-sunward side of Mars. Mars's obliquity is ~25 degrees, just a bit larger than Earth's, and that obliquity does not rotate with Mars's revolution around the Sun. For part of a Martian year the pole would be oriented more or less toward the pole-sitter, yielding a good illumination geometry as shown in Fig. 2. But half a Martian year later the pole would be more oriented toward the Sun, and the pole-sitter would see the polar region close to (or even beyond) the limb of the planet, as shown in Fig. 3. (Oops. I just noticed I forgot to switch directions of the "Sunlight" arrow in Fig. 3. It should point to the left, not the right.) This makes for a poor, or even impossible, illumination geometry. The energy input for vaporization might be very seasonal.
There would be several trades that would need to be analyzed when designing this system, such as the "one big sail or many smaller sails" trade. I haven't done any of those! (As Gomer Pyle would say: Soo-PRISE, soo-prise!) I'll wait until one of my clients pays me to do that job.
The geothermal option is intriguing. There'd be some large-scale engineering in that as well. Would there be geologic hazard implications of cooling a fair chunk of Mars's mantle?
You might note I tend to avoid methods that throw large masses of rocks, dirt, dust, and natives into the air, i.e., huge explosions. But there are methods that would involve such, including nuclear explosions within the CO2 reservoir, or impacting a small asteroid or comet onto the reservoir, etc., and those methods have the advantage of getting the project finished in relatively short order—and they also have some distinct disadvantages. (side benefit to the comet: lots of volatiles, including water!) Altering the orbit of a few-km-sized comet or asteroid to an impact might wind up being less expensive than the pole-sitter solution...but you'd better not have anyone living on Mars when you do that!