Does a Jew have to beleive that the Oral law was either fully or partially given on Mount Sinai to be considered to Kosher Jew?
As well as you don't have to believe that the Oral/Rabbinic Law was given on Mt Sinai in order to consider it binding from a Torah perspective. (Or to be a considered a Kosher Jew) (See "Lo Tassur" Deut 17)
The reason I feel I have to reiterate something I say in my lectures constantly is mainly because no one else is.... (And the opposite is usually taught)
There is only Torah, Halacha and Agadata...
Believing the first two is what makes you a Jew and third is optional to believe...
And Agadot/Midrashim are the only ones to claim that the Oral law was given at Mt Sinai..
The failure to mention this Judaic fact is what has the Karaite movement increasing daily...
(In terms of Halacha l'Moshe M'Sinai the Rambam limits these to around 30 laws which the Gemara itself debates on many of their points of origin)..