But if you're going to insist on Co^te d'Ivoire rather than Ivory Coast why not insist on the correct tonal pronunciation for Beijing?

Well exactly. Syllabic tone is unnatural in English so shouldn't be used when speaking English. One good principle is: if possible (which includes "if you know them") use the closest comfortable English sounds to the foreign sounds. No-one should be expected to know or reproduce Chinese tones.

The case of Co^te d'Ivoire is entirely different. This used to be called Ivory Coast in English, then the English name of the country was changed by explicit government decree, and the new name has accordingly been taken up by official bodies such as the UN. What better or clearer reason could there be for you to use the new name of something?