To me, Beijing is merely the way the Chinese chose to latinize the Chinese name of the city. Moskva would be a way to latinize the Russian name of that city. In French, I see no more reasons to use Beijing or Moskva instead of Pékin and Moscou as to say Lisboa instead of Lisbonne or London instead of Londres.A comment about "Beijing" and other Chinese names.
This spelling dates from the late 1970s, when the Chinese government, tired of seeing every nation spelling Chinese names in its own way, decided to establish a standard international spelling in the Latin alphabet.
Although this contained some very strange phonetics (for example the "ch" sound is now represented by a "q"), it was adopted by the English-speaking countries, who were used to illogical spellings in their own language. Mao Tse-Tung became Mao Zedong, Teng Hsiao-Ping became Deng Xiaoping and so on.
I don't know if any other countries adopted the system, but it had no success in Italy. Very few Italians would recognise Beijing as the capital of China and even less would they know how to pronounce it. Older names, written with English phonetics, are simply pronounced all'italiana: Hong Kong is pronounced without the H and with the final G separate from the N.
The point here is that Beijing and Nanjing are not changes of name, but an attempt to establish an international standard for the existing ones. But if Europeans do not even learn the phonetics of each other's languages, why expect them to do it with Chinese? This is why the attempt failed.