Why does anyone believe in “invoice prices” in the first place?
Or believe that invoice == real cost?

If you look at a big publicly traded traded auto dealership conglomerate, AutoNation (symbol AN), and look up their financials, you will see that for the last listed quarter their Gross margin was

% 425/3127
0.135913015669971

or 13.6%. Does it seem reasonable that car dealers have 13.6% gross profit at the same time as they are selling cars for little more than “invoice” in many cases?

An invoice may very well be an “invoice” of sorts, but I seriously doubt that it represents what the dealer really pays for the car.