Why not ベジン or even 北京｛きたきょう｝?
Supplementing the existing answer. Short answer: because there's neither /b/ nor /j/ in that word. Nor do they even exist in modern standard Chinese to begin with.
This may come as a surprise, but modern standard Chinese, aka Mandarin, lacks most voiced consonants. So there's no /b/, /g/, /j/, or /d/. Although the official romanization system for modern standard Chinese, aka pinyin, prescribes these phonetic symbols, they are really realized as non-aspirated voiceless consonants. But in Japanese—and English I may add, as well as most European languages that I have knowledge of—voiced consonants are prevalent.
The way Beijing is actually pronounced is something close to /peiching/.
For more accurate and technical explanations, check out this Wiki page: https://en.wikipedia.org/wiki/Standard_Chinese_phonology
Technically, the sound that the pinyin j represents is [t͡ɕ], the voiceless alveolo-palatal sibilant affricate, and it's the same consonant as /ch/ in ち.
The sound that the pinyin b represents is just a good ol unaspirated /p/, as the /p/ in spy.
This is actually the reason why native Chinese speakers seem to have a hard time telling apart か and が, ぱ and ば, た and だ, etc. See, for instance, this Q&A:
It came from a reading that existed in China. Note 北京大学 is Peking University.
PekingPékin is used to call 北京 in French as well (like ペカン).