Can I swim in chlorinated water if my nail is coming off?

Patient: I dropped a shelf on my big toe and now it’s blue around the nail and the bottom left half of my nail is coming off. I teach swim lessons for up to 4 hours and I was wondering if I should be going into chlorinated water for that long when I don’t have any waterproof protection for my big toe. Also, can I cut off the part that’s coming off? How long will it take for the nail to fully grow back?

Symptoms: Discomfort, I want to rip it off, area around nail is sensitive