I see, sorry about that, I thought you meant for $S$ to be a subset of edges.

It is a theorem of Haxell that an independent transversal exists in general when $\Delta(G)=k$ and $|V_i| = 2k$, (see here link.springer.com/article/10.1007/BF01793010 or here link.springer.com/article/10.1007/s00493-007-2086-y). By the way assumption (3) would be necessary. Take three disjoint copies of $C_4$ and colour the vertices in circular order 1,2,1,2 and 3,4,3,4 and 1,3,2,4 respectively. This graph has no independent set meeting all four colours.