Premise:
The cat's out of the bag: The Amputator is utter garbage.

I feel like it needs some difference in kind, as opposed to just difference in scale. When everything is defined by scale (i.e. comparing Acheron to Amputator), there's no really good way of balancing them without giving Amputator its own damage type, but it's unique in that it is the only normal damage Brandish.

So I feel that it needs something to make it feel unique. Even if it still is worse than Acheron, so long as it has some niche use, something to set it apart in a way that numbers alone can't easily replicate, then, in my opinion, this is a great weapon. It's niche weapons like this that allow players who have exhausted all the "good" options have something new to play with. A great example of this is FoV; it's not really that good, but it's certainly unique. (Okay, it's a boss weapon so it's relatively easy to get, but...I think my point still stands.)

Of course, whatever this "special" thing is, it can't be too insane to code. I know Grey Havens is a busy company.

The idea:
Give the Amputator: "Slime Execution Chance: High" instead of the damage bonus it currently has.

What is Execution Chance? Well, I'll define it as: "For each level of Execution Chance, each hit on the given monster family has a 2% chance of taking lethal damage." So, in this case, every hit would have a 6% chance of immediately destroying Slimes. This translates to roughly every 17th hit having this effect.

What does this do, gameplay-wise? It means that you can Change Attack a whole horde of Slimes and some--but not all--of them will poof. This is something no other weapon has. It's not as guaranteed as getting a elemental Brandishes' status procs or Acheron's increased damage, but it still should be noticeable.

I do think, however, that the Jelly King and Queen should both be exempt from Amputator's execution ability. This is far too powerful against a boss whose main defense is their regenerating health pool. I do not think that the giant Oilers in C42 should be exempt, though, and nor should giant Lichens.

Or maybe not. Why not throw a bone to those who bothered to obtain a super niche, 5* weapon and allow them to smear 2 of the game's 9 bosses upon the wall? (I'm not counting Event or Sub-bosses in that, and I'm going off the wiki.) Since they're segmented into three phases each, you have to get 3 executions to clear the boss.