Like.I have had success along those lines with something like this Google pic....
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I just used some 2X2's for the frame and tacked on some chicken wire. It keeps the rabbits and deer out of the raised bed. Most of my raised beds are 4X4 feet, so no big deal to lift it off if/when I need to. However, most of the time, the cage sits on top of the raised bed for weeks, if not months, after I set it up.
If you have longer, bigger, raised beds, you could attach the cage frame to the raised bed with some hinges and prop it up with a board when you need to get inside.
What I like about the lift off cage approach is that the same cage will fit on any of my other 4X4 raised beds. I often move my plants from year to year into different beds, so the cage just get puts on the bed where I need to protect those plants.
I could see that if you have a larger raised bed, maybe some heavier, more permanent 2X4 framing with removable panels would work better. In any case, it looks like it might be an excellent use of pallet wood 2X4's or other reclaimed lumber.
The deer have never pushed off one of my cages to get to the plants, but if they did, I'd just add some hinges and/or hooks to lock the cage down on to the raised bed.
I know in my case, it was much easier and more cost effective to protect my raised beds with a chicken wire cage where needed instead of trying to put up fencing to protect the entire garden.
I wonder if a cattle panel with bird netting over it might work?