Suppose, if possible, no such triangles exist, and consider the parity of the total number of pairs $(T, e)$, where $T$ is an isosceles triangle, and $e$ is an edge of $T$ whose endpoints have different colors.

On the one hand, under the above assumption, the vertices of a triangle in a pair form a bichromatic set, so the triangle occurs in exactly two pairs, and the total number of pairs is therefore even.

On the other hand, if we fix an edge $e$ whose endpoints have different colors, the number of isosceles triangles matching $e$ is either one or three. It is one if $e$ is the edge of an equilateral triangle. Otherwise, it is three: one triangle with apex at one endpoint of $e$, another triangle with apex at the other endpoint of $e$, and one more triangle with apex opposite $e$, since $n$ is odd. Since the total number of bichromatic edges is odd, so is the total number of pairs, contradicting the parity established in the previous paragraph.