What is the proper way for a fasting person to rinse the mouth and nose during ghusl?

It is obligatory to rinse the mouth and nose during ghusl. Swishing water thoroughly in the mouth and drawing water deeply into the nose is sunnah for those who are not fasting. However, for those who are fasting, it is not appropriate to do so, as there is a risk that water may reach the throat or nasal passage. Therefore, when fasting, one should not be excessive and should rinse the mouth and nose as done during wudu. (Ibn Abidin, Radd al-Muhtar, 1/116, 156)

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu