I find bay-JING, with bay as in bay, jing as in jingle, bay in lower tone, jing in a neutral tone. I wonder if that latter is sort of the tonal equivalent of a schwa.
OK, if you really want to get into it...
Everything beyond this point assumes we're talking about
普通话 or Standard Chinese (Mandarin), and not one of the many regional dialects.
Beijing is written like this:
北京. Two characters, so two syllables. The first character is romanized as
běi and as the tone-mark over the e indicates, it is 3rd tone (the dip-rise tone), but for reasons too technical to get into here, the 3rd tone is not fully voiced and simply dips. The second character is romanized as
jīng, and it is 1st tone (the high-held tone)
not the neutral tone, and emphatically not a schwa. Listen to this
CCTV news-reader and the field reporter, as they say "Beijing" repeatedly in the linked clip.
Having said all that, the tones of Chinese are for Chinese-speakers and CSL students. For English-speakers speaking English, I don't think it is at all necessary to achieve "authentic" native pronunciation, and "Bay-jing" in a normal conversational tone is fine. If you pitch your voice down a little on the first syllable, and up a bit on the second, and say "Bay-jeeng", you'll
really be getting close.