There's "foamed" PVC that comes in 3mm up to 13 mm thick sheets that's used in the sign biz. The "foam" is quite microscopic in size. It has about a 7 year outdoor lifespan. A 4' x 8' of the thickest size sheet in my part of the world (BC, Canada) costs $130 + tax. It mills and carves easily and takes paint well. If you're lucky you can find it in a small range of colours but white is the most common. You can glue it with solvent PVC plumbing pipe cement or other PVC solvents.
Komatex is one brand name, Sintra is another. There's another called Komacel that goes up to 19mm thick ... it's about $150 a sheet. You may be able to get a sign shop near you to supply some, perhaps just a smaller piece to try for a start.