I've always wondered why people are hostile to metal boxes. I guess you think a metal box will shock you and a plastic box will not? That seems like good motivation to get your grounding right.
What a metal box does is a fantastic job of containing both meltdowns from wiring faults, and also assures a breaker trip if a hot wire gets loose in the box (or neutral wire, if GFCI protected). Plastic fails spectacularly at both, and will actually augment a fire, since plastic is, after all, made of petroleum! Yeah, they add borate flame retardant to the plastic, but that only makes it self-extinguish if the external heat source is removed. As long as it's fed by anything else burning, it is exothermic. Not what you want next to plywood!
It seems important to you that this be cord-and-plug connected. I don't fully understand your application, but let's review a few rules.
First, you cannot cord-and-plug connect something that is concealed. The classic case is fluorescent troffers in drop ceilings. There are several other places you cannot.
However, there are also several places/reasons you can. These are covered in NEC 400.7 and 400.8, which you can read about here.
However, fixed in-wall wiring does not play well with plugs. So putting an allowed in-wall wiring method on a plug is a non-starter. You would need to hardwire it, unless your case falls under a flexible cord exception.