Neither. Especially in TX sun. Go fiberglass. Unless you want to be out there every couple of years stripping and recoating.

Western red cedar is the "greener" choice. Most sources are sustainably harvested timber forests in British Columbia. If a part wears out and you need a carpenter to fix it, the repair is going to match better and cost less.

Paint the door with an oil paint neutral (transparent) tint base, with no pigment added, it will provide excellent protection and dry clear. I recommend Hollandlac Brilliant Base Transparent, made by fine paints of europe.