tevel: a mixture containing terumah or an independent issur

The Minchas Chinuch quotes a machlokes Rishonim regarding the obligation to take challah from tevel. Tosfos in Sukkah (35a d”h asya) writes that challah must be taken only if there is enough dough which would be chayav if terumah and ma’aser were subtracted. The R’Sh disagrees; as long as there is a quantity of dough that meets the shiur of challah, even including ter”m which belong to the kohein, the dough is still chayav.

Seems to me that this machlokes hinges on an issue raised by R’ Yosef Engel (Esvan d’Oraysa #2) regarding terumah. Tevel may not be eaten before terumah is separated; once separated, terumah may not be eaten by a zar. Is this issur of terumah just an extension of the old issur of tevel which remains in place viz. a viz the zar, or is it a categorically new issur? One can flip the chakira and raise the same issue with regards to the issur of tevel: is tevel prohibited because it contains terumah, or is it a categorically independent issur?

Tosfos perhaps holds that tevel is prohibited because it is defined as a mixture containing terumah; therefore, the volume of terumah must be subtracted from tevel before evaluating whether it is chayav in challah. The R”Sh, on the other hand, holds that tevel is a categorically independent issur. True, potentially ter”m must be removed from the dough, but that obligation to give a portion to the kohein does not impact ownership of the dough before it is given.