words - Why is there a dagesh in second camocha, but not in first?

In "Mi chamocha ba'elim H', Mi camocha nedar bakodesh" (from the Az Yashir song by the sea, but also between the shma and the amidah in the morning, and elsewhere):

מִֽי-כָמֹ֤כָה בָּֽאֵלִם֙ יְ-ק-וָ֔-ק מִ֥י כָּמֹ֖כָה נֶאְדָּ֣ר בַּקֹּ֑דֶשׁ נוֹרָ֥א תְהִלֹּ֖ת עֹ֥שֵׂה פֶֽלֶא

Why does the second cuf have a dagesh (and thus is pronounced as the harder "C"), while the first has no dagesh (and is thus pronounced as the softer "ch")?

Rabbi Schmuel Mann (of blessed memory) speculated that this is related to how the words were sung, but we had a problem with that because there is nothing special about the trop on the phrase.

Answer

It's one of 10 traditional exceptions to the rules of BeGeD KePeT recorded by the master masorete Ben Asher in his Dikdukei haTa'amim.

Minchat Shai records two homiletic explanations:

1. 
   
   

   The second מי כמכה follows God's name and we don't want it to sound like we are declaring God to be a fellow named מיכה.

   
   


2. 
   

   The stronger form in the latter phrase indicates a strengthening of the Jews' praise.

   
   

The Siddur Tzelota deAvraham finds those reasons weak and proposes two others:

1. 
   

   [I don't understand this one.]

   
   
   


2. 
   

   The song was probably sung in a special manner and the extra Dagesh signifies a subtle pause in the tune which we no longer have. He supports this by noting 4 [5] more of the above referenced exceptions to the rules of BeGeD KePeT are also located here in the Song (גאה [באלים] גאלת כאבן גאה).