I don't think we are talking about that much bleach. Enough to rinse or coat the bowl. My URO didn't even think it was that necessary as he said "If it was that toxic (BCG) he wouldn't let his staff make the mix". In my opinion it is still good to use for maybe the 2nd, 3rd or 4th voids. Just use common sense on keeping the hands clean.
The BCG shortage has been a common problem. Postponing a few weeks isn't a big deal but, if it persists maybe an alternative agent may make sense. See Joea73's response on non responsive BCG. It especially applies with Gemcitabine/Docetaxel intravesical chemotherapy as a substitute:
Suspicious scope worried about non-BCG options - Bladder Cancer Support Forum