The word Cox’s Bazar was a coinage, and has been included as a Bangla word much before. It is totally a Bangla word by now, and not even a lingua franca.
I did not hear any significant movement to change the name. If you heard of any, it could be by a small fundamental group. It will not be effective in Cox’s Bazar district level. Not at all question to be raised at national level.
As far as I observed and found that people of all level despite race, ethnicity, color, political opinion. communal or religious background, are happy with the name.