I see at least three separate issues being confused here.
1-When is it time to reef.
That's a question of both safety AND better boat speed, unless you have seen the polars for your boat or done a lot of testing, most people keep the sails up too long and they actually are LOOSING SPEED TOWARD THE MARK by not reefing early enough.
Which also brings up point #2,
CREW SAFETY. I don't care how young/old/agile you are, if a boat bounces the wrong way and you stuff a counter or table into your ribs, you are breaking ribs. That's just unsafe.
If there are insufficient handholds and you've got a wide fat open boat below--that needs to be looked at. For any sized crew. Now for kids, IF the kid knows what is going on, and IF the captain calls out each tack so there's some warning, all a kid has to do is typically sit down on the floor and they're perfectly safe from tack to tack. It's the ones they don't expect that can get them, and then you need to consider lee cloths and the v-berth or quarterberth and a house rule about being IN it.
Finally, there's the basic rule DON'T ARGUE IN FRONT OF THE KIDS. Whatever the two of you have to do to hash this out, work it out off the boat, and present a united front, because that's the only way it will be accepted with minimal argument by your daughter.
Helmet and PFD down below? I don't think so. If it is that rough, or you are that uncomfortable on the boat, it is time to head home--or stay home, and find some reason to say "Dad's going sailing without us today" without making it into a "because we're not tough enough" inferiority thing for the daughter. Turn it around the other way: "We're going someplace special, and poor daddy can't come with us!"
Kids routinely used to climb trees, fall down, break bones, skin their knees, chip teeth on monkeybars, and no one thought there was anything unusual or wrong about it. You guys may need to just take some time and discuss comfort levels and acceptable risks one day. Someplace quiet, away from home, while there's a babysitter taking care of your daughter.