Question: Is it necessary to gargle the mouth when making Ghusl or will rinsing the mouth suffice?

Answer: When one is taking an obligatory (Fardh) Ghusl, it is necessary (Wajib) to ensure that water reaches the entire mouth. This is a basic requirement for the validity of one’s Ghusl. Furthermore, the act of gargling the mouth is Sunnah which enhances and perfects one’s Ghusal.
(والمبالغة فيهما ) بالغرغرة ، ومجاوزة المارن ( لغير الصاءم) شامي ج١ص١١٦
(غسل ) كل (فمه) شامى ج١ ص ١٥١
And Allah Ta’ala Knows Best