The angle $168^\circ$ is constructible
Solution 1:
Yes it is constructible.
The canonical pentagon is indeed constructible, as mentioned by the OP, and so is its angle $$ \varphi=\frac{3\pi}{5}=\frac{540^\circ}{5}=108^\circ $$ Also $60^\circ$ is the angle of equilateral triangle, and so it is another constructible angle.
Hence $\vartheta=108^\circ+60^\circ=168^\circ$ is also constructible.
Solution 2:
The 5gon and 6gon are constructible, hence so are angles $72^\circ$ and $60^\circ$ Note that $168^\circ=14\cdot(72^\circ-60^\circ)$.