I'm very sorry to hear that your DD was injured, & that you're being faced with this decision about your dog.
Yes, if he's not real stable, & requires constant upkeep, I'd put him down. It's extra hard since he's been your working companion, but there are stable dogs out there, & you've done a lot with him, but he can't be trusted around the human "puppy" (I had a dog who wasn't good around pups (& developed jealousy issues around other family members regarding his possessiveness of me), & he ended up neutered despite a Ch., great OFAs, CGC, & TT - fortunately, by the time my DDs came along, he was hitting old age, & he died when my 1st daughter had just started walking). Until recently, we also had a 15 1/2 year old Staffy Bullx, & while we kept an eye on the girls that they didn't pester him, he was fool proof, had no food issues whatsoever, & was the perfect family dog.
This isn't just a glitch in an otherwise stable dog - this is the last straw, & you don't want to pass a problem dog off on someone else.
I love my remaining dog dearly, but if he wasn't entirely amiable around my children, he'd be re-homed, pronto.