Let me give an answer from a slightly different point of view.

Let $M_k$ be a moduli space as in your question; say it's a (compact) moduli space of sheaves on some (compact) Calabi-Yau threefold. In general, $M_k$ is going to be very singular. However, it carries a so-called perfect deformation-obstruction theory of dimension zero. This gives a virtual fundamental class on $M_k$, and the technical definition of the numerical invariant $e(M_k)$ is that it is the degree of this virtual fundamental class.

In the case of sheaves on CY3s, the deformation-obstruction theory has a duality property: it is a symmetric obstruction theory. In this case, according to a result of Kai Behrend, $e(M_k)$ can also be expressed as an Euler characteristic, albeit a weighted one:

$e(M_k)=\chi(M_k, \nu_{M_k})$,
where $\nu_{M_k}$ is the Behrend function of the singular space $M_k$. In other words, one computes an Euler characteristic, but weighted with a numerical measure of how bad the singularities are.

On can hope that this Euler characteristic definition can now be turned into something motivic. What one needs is a way to attach a motivic weight to points of $M_k$. In some specific moduli problems, such as for Hilbert schemes of points where at least locally the moduli space can be expressed as a critical locus of a function on a smooth variety, this can be done using the tool of the motivic vanishing cycle; indeed, this is what our work does in the paper you cite. The general theory of how one attaches motivic weights is discussed in a (partially conjectural) paper of Kontsevich and Soibelman.

The issue with Gromov-Witten theory on a CY3 is that the deformation-obstruction theory in that case, while it is of dimension zero, is not fully symmetric. It is symmetric on the open part corresponding to stable maps which are immersions from a smooth curve, but (as an expert assures me) not on the whole moduli space.