You wouldn't want to drill through the trailer door. That will cause you problems in the long term.
Typically, when you screw a fastener into a metal wall/door of a trailer you do not penetrate all the way through. If you do go through, you would only do so on an interior wall making sure to cover the protruding end of the screw.
Find a trailer "store" nearby, preferably one that deals in horse trailers. They will probably install the hanger for a very small charge or at least give you some tips on how best to do it yourself.
Again, do NOT drill a hole straight through your door.