Is local non-satiation enough to talk about Walrasian Equilibrium being subset of Core
Yes, local-non satiation is sufficient for the result. In fact, the theorem holds as long as for every consumer every optimal consumption exhausts the budget. The wikipedia page has a proof which uses local non-satiation: https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics#Proof_of_the_first_theorem.

As a side note: Cobb-Douglas preferences do satisfy strict monotonicity on $\mathbb{R}^{n}_{++}$. Thus, if you can argue that you may restrict attention to bundles where all entries are strictly positive, the arguments based on strict monotonicity applies.