Can I use fluorescence microscopy with a solution of hydrogen peroxide?
The Hydrogen peroxide will definitively accelerate bleaching of your fluorescent microspheres. In fact, we routinely add glucose oxidase, glucose and catalase to our single molecule fluorescence experiments in order to get rid of oxygen and hydrogen peroxide in our solution and thus reduce photobleaching. However, Fluorescence microspheres are very photostable and whether or not they work for your application depends on several factors:


how long the microspheres have been in the solution before you start imaging (the sooner you start the better, because of less reaction time between dye and hydrogen peroxide). 
How long you need to image (the shorter the better, because of less photobleaching)
How large the microspheres can be (the larger the better because large microsphere contain more dye that is less accessible to the hydrogen peroxide in solution).


Basically, I suggest you try (and let us know how it went :) )