The problem is, cauterizing a wound implies tissue damage to the extent of closing off blood vessels. Such damage leaves a layer of dead tissue on the top. And the dead tissue will slough off in a day or two and take the metal with it. Yes there are metals that melt at temperatures that won't kill the tissue. But they won't cauterize the wound.
So it's very unlikely that molten metal will both stick to the wound and cauterize it.
It's just barely possible to prepare tissue for accepting metal and keeping it there. You would have to get some portion of it sub-cutaneous, so that as the tissue grew it would not eject the metal. Basically, you are doing the equivalent of a little anchor in the flesh. Probably it would be difficult to make this stable and long lasting, unless you put it quite deep. Skin is capable of gradually ejecting foreign material unless it's really well embedded. Example from Live Science.
If the splinter isn't removed, the body probably won't absorb the
invader or break it down. Rather, the body will likely try to push the
splinter out, Biehler said.
Cauterizing is usually a "battle field" type operation, when the bleeding has to be stopped and only rudimentary medical capability if available. It's not what would be done in a hospital, for example. So preparatory equipment to make the tissue accept the metal is not likely to be standard in a battle field med kit.
So the combination of cauterizing and keeping the metal in there is extremely unlikely.
However, who says it must be combined? The metal could be added after the wound is at least partly healed. For example, to make the wound look less gruesome. Or more gruesome, depending. A shiny metal covered scar is probably more difficult to miss. Or possibly as a badge of honor or symbol of combat veteran status. Or various other possible intentions, depending on the culture and context. The notion of cauterizing the wound with molten metal could be the popular lore to make the fighters even more scary.