I was actually thinking the top face would be left unglued then you can grip the lasered edges of the top and lift it to open.
Pretty sure I can split the 3D model so it has mating top and bottom sections, while maintaining the parametric design. Then you would just need to assemble the pieces and hold the top and bottom parts together to add hinge hardware to the back, on the outside of the box like a chest or jewelry box.
If you want to do an interlocking lip, it might be easier to cut a matching ledge on the laser out of the same material, that can fit right into the opening. Or glue squares into the front corners and use a hobby knife to cut a mitred chamfer so when the top closes down, the chamfers help seat/align the top properly.
I gave a brief run-through of the workflow for producing a vector PDF file directly from Fusion 360, you can import into Illustrator. It’s at this link: Parametric Box with Finger Joints - Fusion 360 file - #44 by mpipes