Dave,
My primitive thoughts on the matter are this:
If the two spinning masses are on one shaft, then they cannot both be tilting in different directions. That's the point, really. Sure it takes force to tilt one gyroscope, but if if a second gyroscope acts to prevent that motion and there is therefore no total tilting motion, then no work has been done to change the orbit of either gyroscope.
If you do tilt the object as a whole, whatever resistance is offered by one gyro is always nullified by the other and only the static mass properties remain.
To take this to an extreme, think of billions of gyroscopes assembled into one structure. The spin of each gyro is random, so the total angular momentum is zero.
That pretty much describes ordinary matter, which is made of atoms, each of which is a little gyroscope.
By your reasoning, all matter should resist being moved about any axis because all the gyro forces of all the atoms somehow add together.
Chuck's experiment with a drill and some odds and ends confirmed it very nicely. Only then did he dine on crow...