Lining inside a plastic enclosure sounds awful. Wrapping the outside though, that has some promise. Use EMI tape, the kind that makes contact through the adhesive side, or if nothing else, solder over the seams.
The plastic being on the inside, has a minor effect on resonant frequency. Choose plastics with low losses, i.e. avoid PVC and PET, most hydrocarbons are okay.
Can also solder together sheets of copper clad PCB, again doing up the seams fully. Can't exactly get a soldering iron down inside a shape, you'll have to do it from the outside which means the inside copper is kinda just dead space. Might rather use single-sided, actually? Maybe glue up the seams for strength, then solder foil around the corners? Oh, or solder foil around the edges of double-sided, then solder it up the rest of the way.
Tim