Well, the cup in the verse I quoted is a metaphorically drink. It's saying we should not compete for the world (dunya) or women or wealth or what not, but rather for this drink. Now it would not make sense if we got the whole drink, then competition doesn't make sense either. So it's partial of this drink which the final end is Musk. This metaphorical spiritual language and recall the drink talked about in Du'a Nudba to be given by hands of Imam Mahdi (a) himself.
The Kawthar is a reality that unites the next world, Mohammad (s) and God and why God condemns anyone who desires the life of this world to hell.
At the end, it's saying this what we should compete for. But as we all have different capacities to drink from this cup, it uses a kasra, but this is the final end of the drink, the journey, the blessings are complete and the light is complete. Of course journey of God continues forever, but each gains from drink to their capacity.
Now Mohammad (s) being final of Nabiyeen, you would not use kasra, but fatha, because it's not part of Mohammad (s) that is the final of the Nabiyeen.
But you were asking for finality seal type meaning, and here it is. It's the same word.
I am saying, Quran is Arabic.
If the word Khaatam means Last, then in Arab language there must be some examples where people in their daily conversations or poetry or whatever else, use that word, with the intention to mean Last.
So, if Khaatam means last, why we cannot find an example in Arab language where it means Last?
For example, let's say you want to say,
The Last king of Persia was Pahlavi.
If you say this in Arabic, can you use the word Khaatam?
Or for example, you want to say, Last day of the week is Friday. Can you say this sentence in Arabic with Khaatam?
Why not? Why we cannot fond another example where this exact form Khaatam, means last?
I think it is a fair question, don't you think so?