We all know the Cautery Sword is just about the worst choice you can make when upgrading the Brandish, so why not add an effect or advantage to make up for it's overall shortcomings?

Some ideas are

HP drain the opponent
Heal allies
(You know, cautery = medical tool)

Slight chance of burn
Or shock
(because of the "heated blade", there must be electricity also)

Further alchemy path
Increase health

Thanks, I'd look forward to seeing it, if ever.