As it stands now the Cautery Sword line doesn't really compete with it's sibling lines. A large part of this is because damage boosts can't go past +6 or Maximum, but also because even when considering Neutral damage it is outdone by the Combuster...
So as it stands I'd like to suggest a bit of a change to the line, rather than keeping everything the same from the Cautery Sword and just boosting the damage as you craft it upward... how about the idea of the weapon evolves?
Starting out the Cautery Sword is only strong enough to pose a threat to slimes and as such it should keep the Slime damage bonus...
The Advanced Cautery Sword on the other hand gets a boost to it's cauterizing power, changing from a Slime damage bonus to a Minor Fire status, and this can be inflicted on the 3rd swing of the Combo like the Hot/Searing Edge, as well as the Charge attack
For the Amputator you can go even further with this idea of it causing wounds that can't be healed and ADD a second status with Minor Poison on the Charge attack alongside Minor Fire....
While I'm certain that this wouldn't let it outperform the other Brandish lines, it suddenly goes from being burdened with a useless perk to being an all around useful utility weapon, capable of being used to deny healing and deal damage over time to enemies.
As someone who actively uses Amputator I would not like to see the Fire status effect on the Cautery line added.
I specifically pick Amputator in situations where I want a general Brandish-type sword that lets enemies ride the full wave without setting them on fire - so that nasty enemies still can be frozen with a Shivamist any time.
If Amputator had Fire, it would make dealing with oily and toxoil slines much more annoying and it leaves Acheron as the only Brandish that lets enemies ride the full wave without denying frosting.
Now if we talk about status effect, then adding Stun could work. And Poision would also be neat, although I don't think it would fit this sword's thematical nature.
Some sort of healing boon would also be nice and absolutely in line with its thematic. While I don't think that straight up healing other players would be a good idea due to exploits, it could have a significantly increased heart drop rate of enemies killed with this sword, similar to Seraphynxs. Or talking about Seraphynxs, the cautery line could actually have a small chance to actually apply that extra-heart-drop-rate buff that Sera's can do via charged attacks.
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------ So much about status effects ------
On the other side there is the straight up damage difference between normal damage type weapons and specialized damage types in general.
I never understood, why a specialized weapon deals more neutral damage than a normal damage type weapon of the same type.
Like Spupernova < Polaris. (Against jellies or demons)
Iron Slug < Winter Grave. (Against constructs or beasts)
Amputator < Archeron. (Against constructs or beasts)
Valiance < any of the specialized blasters.
etc etc etc.
If a player wants the most damage against an enemy, then he will already bring a specialized damage weapon along. But additionally it also means that this specialized damage weapon out-damages their neutral counterpart. In other words, the neutral counterpart is only better in 1 out of 3 cases, when fighting against a monster family that the specialized weapon is weak against. But for that case it is enough if players just bring another specialized weapon that deals at least neutral damage against that enemy type, which combined instantly covers everything that normal damage type weapons could ever hope to compete with.
Or in other words, here is what I suggest:
Have neutral damage type weapons deal more neutral damage than their specialized damage counterparts.
Then specialized damage weapons still retain their value as the top damage weapons, but neutral damage weapons now perform better in 2 out of 3 cases, (but not with the highest damage spot) and thus bring them in a better spot than what we have now.
It gives neutral damage type weapon their respective position as allrounders and make specialized damage type weapons... well ,,specialized" after all, instead of the specialized damage type allrounders, what we currently have now.