I'm not a fan of converted bird cages or the DCN (Double Critter Nation) cage (the one shown with the chinchillas). We have a DCN (used for rats and later for birds) and they simply aren't appropriate for rabbits. The base is too small -- same for bird cages. (Those ramps are also useless for rabbits.)
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@Poopy Poo , of the photos you posted, only the dog crates and the last cube cage would be appropriate for rabbits. The bird cages, rat cages, and one-grid tall guinea pig cages are not appropriate.]
Multi-levels is fine but the base (footprint) needs to be reasonably large (like the 2 grid by 4 grid cage in last photo). Ideally, the bottom floor should sit on the floor of your home, not raised on a stand. The reason for this is something you've already experienced -- having to pick bunny up to get them in and out of the cage. This is not ideal and in fact can sabotage your attempts at bonding with your rabbit.
The easiest thing for your to do is have a cage that sits on the floor so that exercise time simply means opening the cage door. You should begin with an exercise pen that can be set up around that opening so that bunny's roaming space is limited. This set-up has multiple advantages. One is that you no longer have to chase (and scare) bunny around to catch him and put him back in. All that is needed is to shrink in the x-pen until he has no option but to hop back in his cage.
The other advantage is that bunny has constant access back to his cage -- this alone has a few advantages. He can retreat to his safe zone (his cage) at will. Knowing he can retreat at will can provide a great deal of security & comfort. 3.) He has access back to a litter box as needed which will be most useful if you intend to have him free roam.
I believe the pages on my website about housing, bonding, and introducing a dog and rabbit will be most helpful to you. Normally I may link a specific page, but in this case I'll give my home page
here. If you can view it on a laptop or computer, it is more user friendly than a mobile device.