He may not completely understand rock climbing but he's not wrong that it involves flexing, torquing, and stressing your foot in all sorts of unusual ways. You're probably safe to climb normally when the bone is healed, and your Dr. can tell you when that is.
In the meantime, did he give you a boot to wear? Those are designed to keep your foot totally flat while you walk without flexing, but rather rolling the sole of the boot. I wouldn't trade that out for running shoes until it's healed. When I broke my foot I climbed with one leg, kept the boot on for protection and used a knee pad (the soft kind volleyball players wear, not the hard plastic kind) to protect my knee when it would brace or even smear on the wall. All with a TR belay kept VERY tight while close to the ground. It was a fun challenge, kept me mostly in shape, and forced me to climb more dynamically which was a good thing for me.
Later when it was half to mostly healed, my very climbing knowledgeable PT cleared me to ski (but no ice climbing) due to the stiff boot and low stress on the foot.
Good luck finding ways to cope and stay active, but don't risk prolonging your healing time. Future Stephanie will thank you.