This is Kahn and Sujatha's category of birational motives. The main references for it are https://webusers.imj-prg.fr/~bruno.kahn/preprints/birat-tri-imrn-pre.pdf and section 5 of (the already published) https://arxiv.org/abs/1304.6059. As a consequence of the general weight structure formalism, the Grothendieck group of this category is isomorphic to that of the (additive) heart of the Chow weight structure for it; the latter is the category $\operatorname{Chow}^o$ of birational Chow motives (as described by Kahn and Sujatha; you have to invert the characteristic $p$ of $k$ in the coefficient ring if $p>0$ to deduce this fact from Gabber's resolution of singularities). This means that the class of $C$ is that of its weight complex $\in K^b(\operatorname{Chow}^o)$ (see Proposition 4.1.3(9) of our paper for a general concise description of properties of weight complexes and sections 3 and 6 of my https://arxiv.org/pdf/0704.4003.pdf for more detail). Thus you can use any additive functor from birational Chow motives for your calculations; this functor may be represented by a (birational) Chow motif.

The idempotent completion of you localization is Kahn and Sujatha's category of birational motives. The main references for the latter are https://webusers.imj-prg.fr/~bruno.kahn/preprints/birat-tri-imrn-pre.pdf and section 5 of (the already published) https://arxiv.org/abs/1304.6059. As a consequence of the general weight structure formalism, the Grothendieck group of this category is isomorphic to $K^0$ of the (additive) heart of the Chow weight structure for it; the latter is the category $\operatorname{Chow}^o$ of birational Chow motives (as described by Kahn and Sujatha; you have to invert the characteristic $p$ of $k$ in the coefficient ring if $p>0$ to deduce this fact from Gabber's resolution of singularities). This means that the class of $C$ is that of its weight complex $t(C)\in K^b(\operatorname{Chow}^o)$ (see Proposition 4.1.3(9) of our paper for a general concise description of properties of weight complexes and sections 3, 5, and 6 of my https://arxiv.org/pdf/0704.4003.pdf for more detail). Thus you can use any additive functor from birational Chow motives for your calculations (and compute the class of the corresponding complex that is certainly the alternated sum of classes of terms); this functor may be represented by a (birational) Chow motif. However, studying "classical" realizations (in particular, the "birational Hodge" one) seems to be reasonable also.