1. Use wood fill/patch and hope it accepts the stain like the plywood.
2. Fill it and install aluminum trim with a 1/8" lip in the door frame. This may be a problem as the corners are not square, they have a 1" radius.
3. Fill it and paint the outside of the trailer. Consider it a learning experience, live with it and pay closer attention on the next one.
Here's a picture of the gouge.
The gouge is located along the back edge (not the hinge side) of the door opening about 3" below the corner. Look for the small blue piece of tape.