[Christ]

That might work by just using a few pieces of heavy wire, making a "cradle" for the rock to sit in, and making a base that's wider than the cradle to sit in the ash at the bottom of the box.

Someone above mentioned pre-heating, and that's not a bad idea at all. You could use part of the heat from the exhaust vent to pre-heat the oil, which will make it quicker to combust and cleaner burning. To that end, you may even be able to drip the fuel slowly onto a plate or screen directly, and allow the flames to vaporize and burn it.

Most fatty oils (and several petro oils) will expand away from a heat source, thinning proportionately, so if you allow already hot oil to drip on a flame-licked plate, it should vaporize and burn quickly.

You'll need to adjust your mass intake accordingly, if it's automatically controlled.