I like Hugh's suggestion of using all-acrylic materials, but share your concern about drying. A water-borne adhesive would eventually dry, but could support mold/mildew in the meantime. A solvent-based adhesive would dry more quickly, but may react with the paint, or the acrylic sheet, or both.
Rebecca eluded to another problem with her comment about shearing of the paint layer. And that's a problem you would face with ANYthing bonded to the back of the paint layer.
Acrylic expands & contracts with temperature changes. When your art "sandwich" is hanging on the wall, its face (in open air) would react more quickly to temperature changes than its back (against the wall).
When one side expands/contracts more quickly than the other, the result is a sort of rubbing action between the sheets, which could shear the paint layer. Yes, it is slight -- but constant -- and could severely damage the paint layer over time.
Of course your paint layer should be protected, but gluing anything directly to it is probably not a good answer.
How about having a shallow acrylic box built? You could have an acrylic fabricator miter the edges of your face-acrylic sheet, and bond acrylic "shadowbox" strips of about 1/4" to 1/2" depth to it. Then, after your painting is completed, you or he could bond the back-acrylic sheet to it, to finish the box. Only the outer edges would have to be un-painted.
In the event of a future need to disassemble the work, the back-acrylic could be cut away and replaced after paint repairs.
Check the archives for Superior Acrylics, a California company.