I did once have a conversation with someone who pronounced somewhere around Bay-zcheing but he had a rather nasally, throaty accent so I think that may have had something to do with it. Being the mimic that I am, I naturally copied him and it probably bled into subsequent usage for a while. Not, I might point out, that I discuss Beijing with any regularity.
As an aside, being a mimic has been very useful in picking up the correct pronounciation of Welsh place names and eastern european surnames.
As an alternative aside, if you ever happen to be in Snowdonia in Wales, keep an eye out for old mileposts and their oddly anglicised spellings of place names. Dolgellau suffers from a multitude of these.