So, it is possible to bond them, but it is a more involved process than just putting them together. Baby bonds are not true bonds, which is unfortunately a common misconception. You are on the right track with getting them neutered. Once they are bonded, they would share all living arrangements. During the bonding process, you would need to supervise them closely on dates in a neutral space (I think this is what you are referring to doing in your bathroom), but separate them when they are not in a bonding session or supervised. The process can take anywhere from a couple weeks to a few months, depending on how much time you have to work with them and how the bunnies get a long.
Most cages are too small for even a single rabbit, dog crates can be a bit better but usually still too small. A lot of people like to use x-pens for housing. They are great for temporary housing as well because you can reconfigure them as needed (like have two pens during bodning, and then once they are bonded just link them together to have one larger pen). You can also make “condos” or cages out of storage cube grids (this is my personal favorite), which can be more customized for the space, and give the option to add second levels or a roof if needed.
So to summarize, I would look into getting 2 x-pens or making pens out of cube grids so they can live side-by-side in your laundry room until they are bonded. Then you can reconfigure as needed if bonding is successful.
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The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.