I'm prepared for Luguiru to come here to tell me that this has been suggested, but here goes. XP

I'm not going to provide names since I suck at those, but a Cautery sword sounds like it's supposed to cauterize (duh), so why not make it a normal damage sword, but remove the useless slime bonus. My solution is simple:

Replace it with a chance of inflicting weak fire on something with its normal combo (same chance as FoV), then (of course) make its charge the same charge as the Combuster, with the exception of the fire being lowered by one grade.

Give it a lower base damage (as it is now) so that it cannot co-perform or outperform the existing specialized brandishes in all situations. It should be slightly worse everywhere because it has the utility of being normal damage.

This is basically what my idea would be like at 5*, then it could be scaled down for 4* and replace 3*.

Novelties: As far as I can remember, there is no pure-normal damage weapon that can inflict fire. Feel free to correct me on this. This would also be the only brandish to inflict a status on the normal swing. With the same swing speed as a brandish, it would possibly be able to used (to a lesser extent) as a general-purpose Fang of Vog, though the fire and damage are weaker, the Cautery line's only advantage being its swing speed and defensive charge (as opposed to the FoV's wildly offensive charge).