You can go probably ahead and let them into the run. You do not have a huge space, so they shouldn't get lost.
Being in the coop may have "encouraged" the behavior, a combination of boredom and slight crowding. Although I'm sure they thought it was space a plenty coming from the battery system. LOL
Try darkening your nestboxes and making sure there is plenty of cushion/bedding. Darkening makes it harder to see the eggs, not quite as tempting to the birds. The cushioning helps accidental breaks, sometimes my hens will accidentally step on and break an egg, once the egg is broken it's free game.
You can also try using whole or half apples and cabbage head. Using large whole pieces of fruits of vegetables gives the birds something to do. Maybe they'll be so anxious to beat each other to the treats, that they'll forget about their eggs.
The battary hens may be a little extra work, but this is the case with most rescues. If you had gotten chicks and raised them up, there still is a chance of getting an egg-eater and that one teaching the others, but because you've raised them you're more attached to them and have quite a bit more invested into them(food, time, shelter, original cost, etc).
I think your fine, don't worry, we'll get it figured out. No worries.
-Kim