On Daf 100a: After the Mishna elaborated on the halachot of a Safek Kohen and added "and we impose on them the restrictions relating to Kohanim and to Israelites - the Gemara asks: what are the additional halachot? and Rav Pappa finds an issue in Menahot.
Why did the Gemara not answer in a halacha relevant to our times: the these Safek Kohen should not accept Pidyon Haben?
In the case that the Gemara brings, the Safek has in a single instance both the Chumra of the Yisrael -- that his Minchah needs a Kemitzah, and the Chumra of a Kohen -- that his Minchah is not eaten. In contrast, there are obviously many rights to which he is not entitled merely because he is not definitely a Kohen, such as Matnas Kehunah which you mentioned, and Nesi'as Kapayim which he obviously must forgo. These stem from the fact that he is not a Kohen Vadai, and not because he has the Chumra of a Yisrael, and therefore do not fit the definition of Chumras Shneihem.
Dov Zupnik