grammar dikduk - Difference in Nikkud for לא כי in Torah and Mishna

The phrase "Lo Ki" occurs in Tanach (Genesis 18, 15):

וַיֹּ֥אמֶר ׀ לֹ֖א כִּ֥י צָחָֽקְתְּ׃

and in Mishna:

רַבִּי יְהוּדָה אוֹמֵר, לֹא כִי, אֶלָּא חֲצִי בָצָל גָּדוֹל (Trumot 2, 5)
אָמַר לוֹ רַבָּן יוֹחָנָן בֶּן זַכַּאי, לֹא כִּי, אֶלָּא כָּל כֹּהֵן שֶׁאֵינוֹ שׁוֹקֵל חוֹטֵא (Shekalim 1, 4)
וְזֶה אוֹמֵר לֹא כִי, אֶלָּא בְסֶלַע לָקָה (Baba Kama 3, 11)

I noted that all Menakedim (patented grammarians) write "כי" in Mishnayot and not "כּי". But in the verse, there is a Daguesh.

Starting the study of "שער אותיות או"י" in Sefer Hamichlol, my astonishment is growing. It seems as if the aleph of Lo is mute and the Kaf need a Daguesh Chazak.

I don't know what the difference is between the verse and the Mishnayot. But I am sure that the menakedim know. I need an explanation.

Answer

I will try to combine translation and cantilation analysis. The answer of @DoubleAA is of hight level. Without his help, I could not find the verse in Melachim and understand. But my conclusion, following Rashi in Vayera and the Targum on Melachim is different. For this reason, let me formulate my own response.

1. 
   

   The first verse in Vayera (Genesis 18, 15):

   
   

   
   

   וַתְּכַחֵ֨שׁ שָׂרָ֧ה | לֵאמֹ֛ר לֹ֥א צָחַ֖קְתִּי כִּ֣י | יָרֵ֑אָה וַיֹּ֥אמֶר | לֹ֖א כִּ֥י צָחָֽקְתְּ:‏

   
   
   

   
   Rashi translates "כי = אלא" INSTEAD in English. and the disjunctive trop Tifcha bellow the "לא", the conjunctive trop bellow the "כי" (mercha) are appropriate to Rashi's translation.Lo; ki tsachakt. No, instead you did laugh.


2. 
   

   A second verse in Vayera

   
   

   
   

   וַיֹּֽאמְר֣וּ לֹּ֔א כִּ֥י בָֽרְח֖וֹב נָלִֽין: ‏

   
   

   
   is similar.
   

   The verse in Melachim I 3, 22

   
   

   
   
   

   וַתֹּאמֶר֩ הָאִשָּׁ֨ה הָאַחֶ֜רֶת לֹ֣א כִ֗י בְּנִ֤י הַחַי֙ וּבְנֵ֣ךְ הַמֵּ֔ת וְזֹ֤את אֹמֶ֙רֶת֙ לֹ֣א כִ֔י בְּנֵ֥ךְ הַמֵּ֖ת וּבְנִ֣י הֶחָ֑י וַתְּדַבֵּ֖רְנָה לִפְנֵ֥י הַמֶּֽלֶךְ: ‏

   
   

   
   According to the Targum: Lo chen (= RIGHT)! --> You are not right! (It seems that this Chi do not corresponding to one of the 4 translations that Rashi repeated often in Tora's commentary)
   Beni hachai, --> my son is the live. The trop bellow the "לא" is conjunctive (munach) and up to the "כי" is disjunctive (revia). Lo chi; Beni hachai. --> This is not true, my son is the live.


3. 
   

   In Mishnayot

   
   

   
   

   לא כי אלא

   
   

   
   We have the word Ela after chi, so, Ki/ Chi do not be synomim of Ela. consequently the best translation is "right, true". You are not right, instead ... . Therefore the disjunctive trop is bellow the chi: Lo chi; ela.

The Lo chi is a current example of "בגד כפת" after "או'י". The last word is monosyllabic. The word stress is on the Chi. So as in a word bisyllabic Lochi "לאכי" milera, the cantilation of the lamed holam is not ralented, the "כ" is Rafe.
After time, I found in shorashim from RY ben G'anach Ki is Ken or Beemet. here