The pronunciation of any word in any language does not necessarily inform the way it is pronounced as a loanword or by foreign speakers. I wager not a single Korean would recognize Hyundai as rendered by Jeremy Clarkson. Place names are no exception. Delhi, Ontario is pronounced rather differently from Delhi, India, which is not quite like the Hindi to begin with. Neither pronunciation you suggest is much like the Spanish, either Castilian or Mexican.
Most all Spanish names in the American Southwest have been anglicized, the small Spanish-speaking population having been vastly outnumbered by Anglo settlers from the late 19th century onwards. San is invariably /sæn/ not /san/, Los is similarly /lɑs/ or /lɔs/, almost never /los/. Beyond that, the changes are idiosyncratic. San Mateo and the Rodeo in Rodeo Drive retain the Spanish stress pattern, but Los Feliz doesn't. San Pedro, for its part, doesn't sound like either the Spanish or the English pronunciations of the given name Pedro.
Naturally, many competing opinions have been voiced about the pronunciations of various places. For L.A., Steve Harvey of all people had a column in the June 26, 2011 Los Angeles Times entitled "Devil of a time with City of Angels' name":
In the early 1900s, The Times advocated the Spanish version, carrying a box by its editorial page masthead that proclaimed the way to say Los Angeles was Loce AHNG-hayl-ais.… The Times' campaign aside, the United States Board on Geographic Names decreed in 1934 that the name should be Anglicized to Loss AN-ju-less.…
While the Spanish version lapsed into disuse, a debate arose over newcomers using an alternative Anglicized version with a hard G — something along the lines of Loss AN-guh-less.
So, in 1952, Mayor Fletcher Bowron impaneled a jury of experts to determine an official pronunciation, once and for all, for the city.… Loss AN-ju-less.
[/'æn dʒə ləs/]
But official pronouncements are one thing, actual usage is another. In fact, there is no one universally accepted way to pronounce Los Angeles, California. Former LA Mayor Sam Yorty used a "hard G" and "long E" pronunciation (not unlike angle ease); you can also find one or the other or both in many older films and television shows. Surely the pronunciation favored by a longtime mayor and in Hollywood productions cannot be faulted.
The writer Charles Fletcher Lummis apparently had a special dislike of rhyming Angeles with bees, even writing a poem decrying it. But Bugs Bunny (0:07) isn't about to take advice from Charles Fletcher Lummis.
Will M. Cressy reportedly noted in his 1923 History of California that there were 26 ways to pronounce Los Angeles, all wrong.