A partial solution is to spend less time with the physical compass, and do more on the computer. (See my post on Interactive Geometry Software.) This is what I did for years when teaching high school geometry. One way to reduce physical compass use is to use other tools alongside it or instead of it, even when working on paper. For example, copy segments and angles with patty paper, or use Miras for bisecting. In any case, once students master the basics, they can move on to more challenges using dynamic geometry software — see for example my Construction unit.
But I have recently learned that compass and straightedge construction can be done online, in an environment that is not as powerful as GeoGebra, Cabri, or Sketchpad, and not as physically difficult as a physical compass. Thus it is suitable for introductory lessons, and is your ticket out of having to deal with compasses. But of course, you can only do this if your students have access to computers.
And if you are seeking an extra challenge: Science vs. Magic (?) offers a fantastic Web applet to do collapsible-compass and straightedge construction. (A collapsible compass allows you to make circles, but does not remember the radius of the previous circle.) The applet provides specific construction goals, which are fun and challenging, but it can't be used to teach, as you have no control over what is given. The puzzles always start with two given points, and that's it. There is no way to create your own points, so even something as simple as "bisect an arbitrary angle" is not possible. Still, a great and highly recommended enrichment activity.