You can use bleach and the pucks at the same time , you just cannot mix them together (ie putting bleach thru the feeder is a no no).

trichlor puts CYA into the pool and tends to lower ph. As long as you keep those under control , theres no problem using it along with bleach.

The best way to maintain a constant chlorine level (along with everything else ) comes down to regular testing. Staying on top of the conditions is the key