A word for when an arguer states their case again but simultaneously claims the argument should cease?

Solution 1:

Expanding on my original answer as I think there are several phrases which describe this, though none (other than my suggested neologism) is fully adequate. My general feeling is that this is not well described as a being a fallacy but is more of a (fairly ineffective) debating tactic or gambit. It's certainly common, as it indicated by this amusing article about The Last Word.

Of the options I've found, I think the most relevant are:

Special Pleading

I'd say that what the speaker is doing is special pleading, as they are saying that they should be allowed to put their point across but others shouldn't have the right of reply. Special pleading fills your blank nicely, though it admittedly doesn't fully describe the practice in question.

Suppression of Evidence / One-Sided Argument

By preventing further discussion, or attempting to, the speaker is seeking to limit attention to only considerations favourable to their own position. This may introduce a bias in the evidence available to those party to the discussion. It is an attempt to create a one sided-argument. See the Wikipedia entry on Cherry Picking

Appeal to Censorship

I doubt this needs explaining, one source defines this as follows:

Appeal to censorship occurs when dissenting ideas are removed or silenced in order to make a statement appear unanswered, and thus more probably true. The idea comes down to "I'm correct because nobody else disagrees (because I removed [or prevented] all dissent)". RationalWiki

Last-Wordism

It's such a common phenomenon, it deserves a label of it's own. I doubt I'm the first person to use the phrase, but I couldn't find any examples online ... I suggest: Last-wordism

Not quite what you're after, but they are also attempting to convert their argument into a coup de grâce, Which Mirriam-Webster defines as:

• 1: a deathblow or death shot administered to end the suffering of one mortally wounded
• 2: a decisive finishing blow, act, or event - The decision to cut funding is the coup de grâce to the governor's proposal.

Solution 2:

The OP is correct in describing this tactic as an example of "needing to get the last word". It does not provide any additional value to an argument but awards the person the satisfaction of not having to hear a rebuttal. This is inevitably awarded to one party in an "argumentum ad nauseam", but does not add or diminish either side of the argument.

However, claiming victory in an argument based on having the last word, classifies the position as an "argument from ignorance".

argument from ignorance: Assuming that a claim is true because it has not been or cannot be proven false, or vice versa

Solution 3:

But really, let's stop arguing about this is an example of pathos (appeal to emotion). The emotion being appealed to is empathy. An argument should emphasize reason. Whenever you accept a claim based on how it makes you feel without fully analyzing the rationale behind the claim, you are acting on pathos.

"The post quoted above in grey is a good example of Pathos."