After trying several things, I came up with an imperfect solution to framing my 3D kirigami piece. First, I placed the acrylic box onto the piece of glass in a standard frame and glued it down.
I inserted some similar colored paper behind the glass with a quotation by Abram Urban: "My garden of flowers is also my garden of thoughts and dreams." The show is called "Reflections" so I thought this quotation fit the theme.
At first I used a standard glue but it didn't hold so I finally relented and used aquarium sealant glue. This means that if you want to remove the papercut you will have to break the seal on the box. I did not glue the papercut or its backing to anything. It is held together by the pressure of the box on it.
Sheesh! When will I ever learn to consider the mechanics of finishing and framing during the design phase - before I create the work? I am forever barreling ahead into some new piece, and only as an afterthought do I think of how I will display it.