I have seen similar enclosures (not sub boxes, but what the hey) built on boats by "glassing" (mostly epoxying) the outer half to the inner half and the supporting structure.
IE: construct the lower, let it cure, meanwhile build the upper out of the car. When both halves are cured use epxoy and some glass to glass the upper permanently onto the lower and the surrounding body structure. If that will be exposed, you will need to fill, sand, paint just like body work, but if you are carpeting over it you just need it relatively fair.
We use west system epoxy around here because most of us DIY boat people have it around all the time, but there are probably other products, or see www.westsystem.com.
Don't know how much availability there is in Manitoba.....