First, I'm glad you're here to learn more about your new requirement.
Second, a rabbit isn't a hamster you can stick in a cage and look at occasionally. It's not *quite* a dog or a cat, but its much more a member of the family than that. It needs human interaction and attention not just daily, but a couple hours minimum daily. Would you leave a dog in a crate or a cat in a travel carrier all day? I'm assuming no, that the thought is ridiculous and abusive. Well same goes for rabbits. They are amazing, intellegent, loving creatures. They are intuitive and can learn, they'll learn simple tricks, to come when you call, to be litter trained and more.
Here's the set up for my rabbits to see what is acceptable-
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Main things to notice, lots of space, lots of hay, and cardboard. I now have those waterers instead of a bowl, most bunnies prefer to drink from crocks or bowls over water bottles. Its also easier to clean. In addition to their pen, my older bun Samson (you can see her bum behind the air cleaner) is free roam all day, and Lady is still litter learning so she gets her pen and several hours of kitchen time (easier poo cleanup). They also aren't bonded yet as Lady isn't spayed, so I have to keep them separate. Those pens are only 40 bucks on Amazon, you can make a floor if your worried about your floor by just buying some cheap linoleum. But once they are litter trained, they are super super clean (Notice Samsons pen, I scrub down maybe every couple of months with a little vinegar).
They also get nightly salads, romaine lettuce, a little kale and collared greens, and a tiny bit of shredded carrots.
I know you said out time isn't an option because of the dogs and cats, but I think you might be surprised about pet interactions. Is there a way to get a baby gate of block off the other animals in the other room for a hour a couple times a day? Rabbits like to sleep all day and are most active at dawn and dusk, so even out time while you are eating breakfast and dinner would be great. If not, a couple of those xpens would be good enough
I taught Samson to go back into her cage with bribery, she goes in every night around 8 for her salad.
I know you don't want this rabbit, but you've committed yourself to keeping it. You might as well learn about the awesome new addition to your family, and how to give it the best life.
Depending on what that bun has gone through and its temperament, it might be a challenging bond, but I promise its like no other. Start by letting it out more, sitting on the floor (with your laptop?) and basically ignoring him. He'll eventually come up to you, rabbits are extremely curious. When he does, have a treat ready, a sliver of carrot, a craisen, and just hold it in your palm or set it on your leg.
I know I wrote a lot, I'm sorry, I'm just really passionate about this. Probably because when I was younger I had a bunny who I didn't give a good life, and i had no way of knowing that it could've been better.