The following paper of Tim Hodges may be of interest:

$K$-theory of $\mathcal{D}$-modules and primitive factors of enveloping algebras of semisimple Lie algebras. Bull. Sci. Math. (2) 113 (1989), no. 1, 85–88.

He proves that the Quillen $K$-groups of the abelian category of coherent $\mathcal{D}_X$-modules on any smooth complex quasiprojective variety $X$ (such as a flag variety) are the same as the corresponding $K$-groups of $X$ (or equivalently the category of coherent $\mathcal{O}_X$-modules.) The proof proceeds via a reduction to the case when $X$ is affine, and then by considering the associated graded of $\mathcal{D}(X)$.

It follows from this that if $X = G/B$ is a flag variety then $K_0(\mathcal{D}_X)$ is a free abelian group of rank $|W|$, where $W$ is the Weyl group of $G$. It should be possible to construct an explicit set of $|W|$ pairwise non-isomorphic indecomposable projective coherent $\mathcal{D}_X$-modules using Schubert cells.