correct, ajf... beer brite is a cleaner (and one that needs to be rinsed a LOT, from my experience.) I prefer to use just regular old dish soap over BBrite, honestly.
I used to use BBrite or soap to clean and bleach to sanitize, just as you mention. It was a slight pain due all the rinsing that I had to do, but it works just fine and is probably what I would do if I did not have a LHBS.
Also, I agree that you will be very happy if you mail-order a big bottle iodophor or pick one up the next time you happen to be at the not-so-local brewshop.
I use soap and iodophor for everything now.
-walker