I know nothing about your locale's rules of course, but my own experience here in my neck-of-the-woods has been that while commercial activity is often prohibited in residential areas, the opposite is usually not true. That is, residential use of a commercial property is rarely if ever forbidden. The trailer itself might be an issue, but since your locale specifically addresses that in residential areas but not in commercial areas, it likely isn't an issue. (That which isn't expressly forbidden is usually allowed, as they say.)
You MIGHT run afoul of health department/permitting regs or something if there's no already installed sewer connection, but that seems remote.
And frankly: Since it's a car lot anyway, I guess you could just put a "for sale" sign on the trailer. Who's going to know the difference?
All of that having been said, if it was me I would probably check it out with the city in advance. I dislike surprises and wouldn't want to go to the trouble of moving only to find that there's going to be a problem.
ON EDIT Qualifier:
Just want to add that the above presumes that you own the property in question rather than lease/rent it.
