For compact $U^k(1)$ Abelian group, we recently have a paper https://arxiv.org/abs/1906.08270 to define its Chern-Simons theory on spacetime discrete simplicial complex. The quantized coefficient of the $U^k(1)$ Chern-Simons term is given by the $K$-matrix (symmetric integer matrix with even diagonals). Our theory is a bosonic theory and the spacetime does not needs to be spin. So we solved the *Open problem* for compact Abelian group, with an additional bonus: our discrete path integral also has exact 1-symmetry of the Chern-Simons theory.

Our approach does not work for fermionic Chern-Simons theory where
the $K$-matrix is a symmetric integer matrix with some odd diagonals.
Our approach also depends a choice of branch structure of spacetime simplicial complex (which may correspond to the framing structure of continuous spacetime manifold).

https://arxiv.org/abs/1502.00641  defines $\mathbb{R}^k$-group Chern-Simons theory on spatial discrete simplicial complex, with continuous time.