Step 1: Convert obtuse angles to acute ones.
Use $\sin(\pi - \theta) = \sin\theta$: $\sin\frac{8\pi}{9} = \sin\frac{\pi}{9}$, $\sin\frac{7\pi}{9} = \sin\frac{2\pi}{9}$, $\sin\frac{5\pi}{9} = \sin\frac{4\pi}{9}$.
Step 2: Rewrite the product.
So $P = \sin\frac{\pi}{9}\,\sin\frac{2\pi}{9}\,\sin\frac{4\pi}{9}\cdot\sin\frac{2\pi}{3}$.
Step 3: Use the triple-angle product result.
There is a clean identity $\sin\frac{\pi}{9}\,\sin\frac{2\pi}{9}\,\sin\frac{4\pi}{9} = \dfrac{\sqrt3}{8}$ (a standard result for these equally related angles).
Step 4: Evaluate $\sin\frac{2\pi}{3}$.
$\sin\frac{2\pi}{3} = \dfrac{\sqrt3}{2}$.
Step 5: Multiply.
\[ P = \frac{\sqrt3}{8}\cdot\frac{\sqrt3}{2} = \frac{3}{16}. \]
Step 6: State the answer.
So the product is $\dfrac{3}{16}$, which is option (C).
\[ \boxed{\dfrac{3}{16}} \]