Is it that she doesn't take care of them when she is there, or that she is not there half the time, meaning that you have to take responsibility?
I read your OP as meaning the latter - that she does her best but that your separation from her dad means that she can't look after the bunnies half the time. If this is the case, then I think it is really unfair to make her get rid of her pets now. IMO that would lead to resentment on her part, which would be quite justifiable.
You say that you allowed her to get the bunnies, so honestly, once you allow your child a pet, you need to follow through. I don't agree that rehoming animals because they are too much work is morally right, and it would not be something I'd want to model to my children. There are enough homeless animals out there, and to add to that number is wrong imo, unless it is totally unavoidable.
Personally, I'd let her take the new cage, but with the stipulation that there will be no new bunnies.
I'm not sure about how your bunny arrangement works, but we have two bunnies and they get tons of attention. They are in our converted garage, attached to the house, and dds spend hours in there with them. Are yours now neutered? That's the first thing I'd do. Then I'd get them out of cages and let them have more freedom (I'm not sure though about how they'd get along initially, I know they love a companion, but am not sure about setting ten loose!)
Can you build an outdoor run where they can go outside for part of the day and play? Or an indoor area where more than one or two can be out at the same time? Bunnies are social creatures - I"m sure you know that - and maybe rethinking their set-up would make their lives easier and yours too.
Cleaning our bunnies is no big chore. They are litter trained, so I dump litter each evening. The girls - aged 6 and 4 - check water daily, feed hay and pellets, and fresh food. I check it myself each evening or when I pass through the garage, but it's no real 'work'. If we go away, we get a neighbor kid to come in and feed them and empty litter. It costs a minimal amount, so it really doesn't cramp our travelling plans. Can you find a local kid to help out if you want to go away? Or even when you are at home and want a break, when dd is with her dad? How about 4H? Is there a kid there desperate for bunny time who'd want to come and play with the bunnies and learn how to clean them out?
I'm just trying to bounce some ideas around, as I'm sure 10 bunnies are overwhelming! If they are not neutered, that's the first thing I'd do, then get them out of the cages and litter trained so you are not cleaning up ten cages a day! I do one, but ten would be rather too much - sixteen - no way!
Hope something I've said is of some help. Good luck!