Pole Zero Cancellation on a Root Locus
As long as build-ability is not a design requirement, why not place two pairs of conjugate zeroes?  Place one pair next to each pair of conjugate open-loop poles.  Use SISOtool to play around with where exactly the zeroes should be placed in relation to the poles.

If the zeroes are placed appropriately close to the poles, the residues on the closed-loop poles can be made very small, which will minimize the effect of the slow, underdamped dominant closed-loop poles.  (Can you prove this?)

Going beyond (what I assume is) the homework problem, @TimWescott's comment about having a heart-to-heart with the mechanical engineer is a good one.  The closed-loop poles you end up with when you use to above-described "cancellation" method are dangerously close to the Imaginary axis.  What's more, if you were to truly cancel them mathematically, it's extremely unlikely that you would be able to fully cancel them physically.