k disjoint triangles with graph splitting to two distinct groups
Color the vertices in $V \setminus X$ with $2k$ colors at random. With probability $e^{-2k}$, the $V \setminus X$ vertices of your $k$ disjoint triangles will be highlighted.
Suppose without loss of generality that $X = [m]$.
Using dynamic programming, determine for each subset $S \subseteq [2k]$ and $i \in [m]$ whether there exist $|S|/2$ disjoint triangles using the colors in $S$ and vertices $1,\ldots,i$ of $X$. This takes time $O^*(2^{2k})$.