I’m going to assume that you aren’t certified for plastics on the MultiCam? … so here’s another way to do it. All of the steps are fairly simple.
“Rough” cut on laser or table saw - then route using the Incra router - and flame polish to finish it.
If the shelf is rectangular, then the woodshop tablesaw with the plastic blade will be good for the “rough” cut. If you are cutting curves, then laser is clearly the best way - but your workpiece will have to be rough cut on the tablesaw to 63" to fit into Big Thunder. Cut your piece a little larger than you need to allow for routing it.
The best edge will be done using a router after you have made your cuts. Assuming the Incra router (woodshop) is working, that’s your best bet for the fine finish. You can also easily do that on the curves using a bit with a pilot on it. Don’t remove the paper backing until all the routing, etc. is done.
After you’ve routed the edges, you can flame polish them with a torch for a really finished look. Of course, you will want to peel back part of the paper backing before you take a torch to it … and practice a little on a piece of the drop off scrap.
(EDIT: Be forewarned that a piece that large will require a little finesse to laser cut. The power setting required to cut a piece sitting in the rear LH corner of the bed isn’t the same as the power required to cut a piece in the front RH corner. It probably should be, but experience tells us that it isn’t. Your piece will take up the entire bed so this anomaly will be particularly evident. Don’t remove the piece from the laser until you are certain it is cut all the way through and all the way around because you will likely have to make several passes.)