Just say in phonetic English 'meekst' and hold the t a little bit, you will get the ù automaticaly. But sharp prounonciation.
The 'mixte' models became popular in the seventies all over Europe but were originated in France as a utility bike in the countrysite even before WWII. Later in the 50ties and 60ties these elegant bikes became popular among young women in Paris and the Cote d'Azur, south of France. It became a fashionable Lady bike and later in many countries the mixte was built like that. Also the cycling (cyclisme) became more popular for woman, also in racing. Because of the very strong, light and stiff frame the mixed was used as the platform for many woman race bikes.
About Japan, as far as I know, there is no dedicated name for the 'mixte' , in fact all the bikes (unisex) in Japan are lookalikes, I will ask Mihoko, a friend.