I wouldn't focus too much on the origin of the term. They use the word pizza because it's a popular term that would invoke a lot of suspicion.

pizza (n.)
1935, from Italian pizza, originally "cake, tart, pie," of uncertain origin. The 1907 "Vocabolario Etimologico della Lingua Italiana" reports it is said to be from dialectal pinza "clamp" (from Latin pinsere "to pound, stamp"). Klein suggests a connection via loan-translation with Medieval Greek pitta "cake, pie," from Greek pitta "pitch" (cognate with Latin adjective piceus "of pitch"). See also pita.
http://etymonline.com/index.php?search=pizza