To me there’s no question here. High-Density Urethane, aka Sign Foam (in the 15-18#/ft^3 range is what is used for sign making). I think Inventables sells PrecisionBoard in reasonable sizes. A 4x8 sheet, 1.5" thick set me back $400 (~$12/sq ft) but the stuff is amazing! Cuts like MDF, that is to say, disintegrates cleanly, and cuts quickly. Attaches with woodworking screws and the like.
You’ll want to keep this out of the way of serious damage. I suppose with enough force (probably less than a wood sign) it’ll snap, and its soft enough that you can make a gouge in it with your fingernail, but having said that, I made these for a public dog park, and more than two years later, they still look great: