Variants of Cluster-Vertex-Deletion problem
It seems to me that the case where G\X is a disjoint union of exactly two cliques has an FPT algorithm with runtime $O^*(3^k)$.

Here's what the algorithm would do. 


If $k \leq 0$ then output YES if and only if G is a disjoint union of two cliques.
If the input graph contains an induced path on three vertices ($P_3$), then at least one of these three vertices need to be deleted to turn the graph into a cluster graph. Thus we branch in three ways depending on a choice of vertex, decreasing the parameter by 1 in each branch. Output the OR of the recursive calls.
If the graph is induced $P_3$-free, it is a cluster graph. If there is only 1 clique in this cluster graph: output NO. If there are two cliques in this cluster graph: output YES. If there are more than 2 cliques: sum the number of vertices in all but the two largest cliques. If this number is more than k, then output NO (all those vertices need to be deleted to reduce the number of cliques to 2, but we can't afford this). If the sum is at most k, output YES.


The approach generalizes to obtaining a cluster graph with exactly $\ell$ cliques for any value of $\ell$.