No, Peking is not more phonetic than Beijing, and it isn't even Wade-Giles. Beijing is the currently accepted western name. You could make a linguistic argument for Wade-Giles, but not for the postal romanization.
In Portuguese the name is still customarily "Pequim". To illustrate, go to the Beijing page in English Wikipedia and change it to Portuguese. Check what's the title of the page.
That's also the canonical name used in French, Finnish, Italian, German, Spanish, Swedish, and many more. Last time I checked they're all western languages.
Names are chosen by the users. It's not because the English-speaking word was quicker to adopt the new name that it suddenly became "universal". Universal in English? Sure.
I'm sure that eventually most languages will switch to their local spelling of "Beijing", but that is many decades away. You can't change a word that has been used for centuries by decree. Languages don't work that way. Or maybe for English it does, not so much for many others.
In the English language, probably. I'm not a native speaker of English and momentarily forgot that "Peking" is unusual (although – in my defense – the English Wikipedia page still has it). But in my native language (German) everything other than "Peking" would be highly unusual.
