The first recommendation with signs like this is to figure out your mounting before you make the sign. By making the sign and now figuring out the mounting, you've eliminated various options. Simply going to 6mm ACM would have cost you not a lot more but the difference is you can throw a few screws in and they'd actually grab onto something. You could have also drilled holes through the ACM, counter sunk, put in a flat head screw/bolt, and then laminated over it. It's very likely the screw would never been noticeable.
That said, you should be able to mount this safely. Z-Clip/French cleat will likely be the easiest method to mount. The biggest issue with your sign is you've made it in a way that you will have a hard time using a mechanical fastener. VHB tape, adhesive, and some sort of screw that just grabs into the sign even a little bit will probably do the trick. You might find it helpful to cut a few strips of ACM and gluing that to the sign itself first. Then mounting the z-clip with tape, screws, and adhesive. To be perfectly honest with you, if it were me, I'd still probably want to throw a screw or three in there but that's me.