Let us show that there exists a metric for which stable and unstable manifolds of a given Morse function are transverse.

By Kupka-Smale theorem, a Morse function $f$ on a manifold with a Riemmannian metric $m$ can be perturbed to become Morse-Smale, by genericity. The perturbed function $g$ has identical level-set foliation up to diffeomorphism so it is conjugate by a diffeomorphism $\phi$, more precisely $g = u \circ f \circ \phi$, where $u$ is an increasing diffeomorphism on the real line.

 The stable and unstable manifolds of $u^{-1}\circ g$ for $m$ are transverse as composition by $u^{-1}$ preserves this property. Next we apply a global change of coordinate to both $u^{-1}\circ g$ and $m$ using $\phi^{-1}$. This will send $u^{-1}\circ g$ back to $f$ and $m$ to its pull-back by $\phi$. 

Also this change of coordinate will send the stable/unstable manifolds of $u^{-1}\circ g$ for $m$ to the ones of $f$ for the pull-back of $m$. In particular the new stable/unstable manifolds are transverse. So the pull-back metric satisfies the transversality condition.

It remains to address the special Morse condition. It is not difficult to modify $m$ locally around critical points of $f$ so that the modified metric $m'$ is special Morse for $f$, using partitions of unity. We can then repeat the above construction with $m'$ instead of $m$. 

While changes of coordinates preserve the special Morse condition, the perturbation of $f$ might destroy it if it affects neighborhoods of critical points. Fortunately the proof of the genericity of Morse-Smale condition produces a perturbation that does not affect those neighborhoods, so the above construction will satisfy the special Morse condition as well.