I would shoot series of identically exposed shots with tripod, underexposing them enough to not blow the highlights on the flames too much. Then you can stack the set either manually or with programs like Anti-Lamenessing Engine (ALE) to get the surroundings parts properly lit and then use layers and masks to pick flame part from some single frame.

In situation like this, where the subject changes wildly, hefty stack of exposures with identical settings also gives you lots of frames to pick flames from, so you can handpick the version of flames you like most, or even combine it with couple of frames with layers and masks.

Developing with different settings from same raw does not give you any more signal than just converting the same raw to 16-bit image, so there is no real use on doing that, unless you are using photo processing program like GIMP that lacks support for 16-bit images.

Shooting the "normal" HDR stack, ie. set of 3+ images with exposure bracketing has the problem that if subject changes between the shots, like it certainly does here, you easily end up with ghosting. With identical (underexposed) exposures you need more shots to get enough information, but the ghosting in that methods turns to just regular "motion blur", which is more pleasing to human eye.

Also, flames are bright and self-illuminating - if your plan is to capture just the flames, a single exposure might very well be enough. But because memory card space is cheap, shooting couple of series from tripod will leave you with lots of material to choose from.