Switching to bleach is no problem. All chlorine is chlorine once it's in the water. You just cannot mix them together--like pouring bleach over Cal-Hypo powder or Tri-chlor tablets is a really, really bad idea! (don't ask why--just don't do it.)

You can read higher amounts of chlorine (ignore the bromine) with your tester by diluting the pool water with steam distilled water (avail at your supermarket). If you dilute it one to one, you can double the reading you get. Use 2 parts distilled and 1 part pool and you can triple the reading.

I like to use a shot glass to measure because it's simple, easy, and very accurate.