You need enough HCl to dissolve the iron gunk, and dissolve the iron you're adding, AND accommodate the oxidation.
HCl first, then iron. When all copper is precipitated and the iron is dissolving with the production of hydrogen, the solution is completely reduced. Run through a filter and begin oxidizing it. If the solution becomes turbid, add more acid to keep it in solution.
I would recommend using 30% H2O2 because bleach is basic, adds sodium, and reacts quickly, resulting in lots of chlorine gas, even if you stir it in quickly.
Sodium chlorate reacts slowly as well, which can be a problem because you can easily add too much. Although it adds sodium, it doesn't add as much as bleach does. It can sometimes be found as all purpose weed killer, particularly in Europe.
Uh... good idea not to heat a glass container of *anything*. For that matter, even using glass ovenware these days is risky. Pyrex isn't "pyrex" anymore.
Tim