*More comments at end on finding group elements with small word length *
Addendum. Any finitely-generated subgroup of $SO(3)$ generated by matrices with algebraic integer entries (such as this) has a faithful discrete action on a product of spheres, hyperbolic planes, and hyperbolic 3-spaces, by a general construction for algebraic groups, from which it follows that it is either virtually abelian or has exponential growth and in fact it contains a free subgroup. It seems likely that these group elements have orbits on $S^2$ that are reasonably uniformly distributed, although I don't know what's proven about it. If so, by just counting elements it would follow that $P$ can be take to within $\epsilon$ of $Q$ using a word of length linear in $\log(\epsilon)|$.
It looks like an interesting challenge to try to find a polynomial-time algorithm that will find such a word. Once $P$ is within a small negihbrohood of $Q$ and you have a modest selection of moderate-length words that look almost like translations in a magnification of this small neighborhood, one strategy would be to make a first approximation of getting closer by adding vectors. But they are not exactly addition of a vector, and the many different possible orders in which you could multiply them give many different results.If you could systematically analyze and control the effects of changing the orde it might be possible to systematically improve the approximtion. In other words: instead of multiplyingby higher and higher commutators at the end, actually commute elements in the produt.
However, this is more of a challenge than I feel like plunging in to here.