Cute bunny! It looks like you love her a lot, but I have a few suggestions.
Those store bought cages aren’t really big enough for a bunny, especially when she grows. It might seem big to you in terms of the cages they sell in stores, but that is what is wrong with the pet industry.. I have a cage that is the same and we just used it as a “home base” and attached a pen to it.
When you get your xpen you can attach that to the front of the cage, put down something like linoleum as a floor and allow her to live freely in that area. You may need to cover the cage door in carpet or a towel to make it safe for climbling in and out. It’s just so much better and she will have more room for a bigger litterbox (which she might need, I know mine did) and room to run and binky. I don’t know how much you know of rabbits already, but they love to run and do crazy jumps so having a permanently attached pen allows this freedom which is necessary for a happy bunny. I know my rabbits behaviour/mood/interaction improved vastly when given more space to their living arrangements. The minimum is 2ft x 4ft per bunny excluding exercise space for out of cage time.
Your daughter can also sit in the pen and bond with your bunny, and therefore she doesn’t have to be picked up or have anyone reaching into her cage to move her (things that can be detrimental).
I know the pet stores sell things like these cages/corner trays/water bottles and says that’s fine but it really isn’t. At least the cages on their own anyway, some have rabbits that are ok with small boxes and bottles. But these ways are old fashioned and ill informed, rabbits aren’t just bigger hamsters like pet stores have you believe. I switched to water bowls, storage crates for litter boxes and a big NIC condo when I learned this!
When you say you want to put her in the yard, what is that for? Permanent or just for an hour or two? An xpen would be fine just for a bit whilst you were supervising, but I’d never leave it unattended. Any predator can easily get in, even if your bunny is full grown, especially if you know you have owls nearby! But just keep in mind that not all bunnies enjoy being taken outside. I have two and one loves it being outside but the other is absolutely petrified of it.
Good luck with expanding her living space, attaching an xpen to the front of the cage with zip ties and using a linoleum floor is the simplest way to do so. I promise you’ll be doing so much for your bunny just by doing this!
Take a look at the habitats section for ideas https://binkybunny.com/BUNNYINFO/GreatHabitats/tabid/229/Default.aspx