First off, a check with your local building supplier for "148.92 cm x 243.84 cm Expanded PVC Sheeting in 6 mm thickness" is in order. If that fails, then ask about any other smooth surface, paint-able board that is rigid yet light weight . Specifically ask for a material that is used by Sign Makers, as that sort of material is often ready to accept paint without the need for Priming.
Next. A decision as to spray on or roll on the paint is needed. Your stated budget should easily allow for either option, and by a long stretch, spraying will provide you with the most ideal results.
Over viewing your situation, you have a mixed bag. While your room's colors and controlled lighting do not need the help of a Ambient Light screen, the w1070's contrast could use some significant help.
First off, you should ceiling mount the PJ at a distance where the face of the Lens is at 8' -8" (264.16 cm / 2.682 m) to optimize the PJ's light output.
Now. the paint you use should have the ability to enhance / improve the viewed contrast "on screen" without unduly crushing detail or "graying-out" whites. Since ambient light isn't an issue, you have two choices. Either a light, standard neutral Gray paint, or a more advanced paint formula that is still simple and affordable, such as RS-MaxxMudd LL. The latter can be either Rolled or Sprayed.
Beyond those choices, it all becomes pretty simple and easily accommodated. The specific, hard to import paint supplies can be sent to you by a third party, while the basic ingredients can be sourced locally.
You then just mix, strain, and then either Roll or Spray.
As for mounting the board (...and that should be done before painting....) and then trimming the screen (...with Black Velvet...) we can address those specifics once you report on your success (...or lack thereof...) of obtaining a solid substrate material suitable to this application.