Why doesn't 褱 sound like its phonetic etymon 眔? What might explain this phonetic change, in Cantonese and Mandarin? 
Please educate me if there's a more fitting term than 'phonetic etymon'.


Wiktionary avouches the same etymology as Yellowbridge:
Interestingly, there is another character 褢 [same reading as 褱], with 鬼 inside. 

Here are a few parts where Humanum mentions the relationship between 褱/褢/鬼 and even 歸:


  又表示給予，史牆盤：「受(授)牆爾福，褱(懷)祓彔(祿)」，兩句句意相同，意謂授給牆多福。《詩．檜風．匪風》：「懷之好音」，毛亨傳：「懷，歸也。」「歸」有饋贈之義。


So there is a usage of 褱 that means 歸 [kjuj] "give" and related to 饋 [grjujs] "make offering to."


  又通假為「鬼」，伯[冬戈]簋：「隹(唯)用妥(綏)神褱(鬼)，唬(效)前文人」，意謂以安寧神鬼，效法前世有文德的人。


Here's evidence that 褱 [gruj] and 鬼 [kjujʔ] were homophones. Actually, 歸 and 鬼 might be cognates (人所歸爲鬼), but I digress. The point is, [kjuj] is the much more likely target reading to hit for 褱, not the [k-l- | -p] initial/final pattern of 眔.


  「褱」與從「鬼」聲的「褢」字有異文關係，《漢書．外戚傳》：「褢誠秉忠」，顏師古注：「褢，古懷字。」玄應《一切經音義》十八「懷孕」作「褢孕」。


褢 was used in the Han-Tang era to mean "hold in," without the emotional semantics.


  又表示安撫、招來，毛公鼎：「率褱(懷)不廷方」，意謂安撫不來朝覲的方國。陸賈《新語．道基》：「附遠寧近，懷來萬邦。」
  ...
  又借「鬼」為「懷」，表示安撫，《睡虎地秦簡．為吏之道》簡46-47貳：「君鬼(懷)臣忠，父茲(慈)子孝，政之本殹(也)。」


This is the same as the "give" meaning.

We learn several things:


鬼/褢/歸/褱/懷 were near homophones and were interchanged in writing at various times.
There is a "hold in" meaning of 褱 that seems cognate with the "hide tears/miss" meaning for which the character was apparently constructed, possibly a direct extension, but more likely to be the actual root -- there is a Proto-Tibeto-Burman etymon [*kway] that means conceal/hide.
There is a (fairly archaic) "give" meaning of 褱 that is related to some of the meanings of 歸 (namely its use as 饋), and seems like a phonetic loan and unrelated to the first meaning.
The reading of 褱 [gruj] is fairly secure given the readings of the other characters, in particular, the [g/k-r/j-uj] pattern. It is most likely not related to the reading of any crying related etymon, for which there is a fairly established Proto-Sino-Tibetan reconstruction [*krap], nor to the 泣 [khrjəp] that is posited to be 眔, nor to the 徒合切 [dup] spelling given by Shuowen for 眔, nor to 涙 [c-rjuts], 隶 [c-rəts], or 逮 [ləts] that have been written with 眔 in Oracle Bone for its meaning and/or sound.
See also my comment to droooze's answer on why I believe 褱 is a rebus.


So, to answer the question: "Why doesn't 褱 sound like its phonetic etymon 眔?" It's because it has no sound component, and 眔 is not its phonetic part. On the other hand, the later character 褢 would be good candidate if you're looking for a legitimate phonetic part for a cognate of 褱.