When we first got our trailer I thought the plastic might break so I put metal plumber's tape behind it to reinforce it.
I'm sure your plate is languishing in a ditch somewhere with the remnants of the plastic bracket still attached.
When I changed over to LED taillights, I made a new hanger bracket using a scrap piece of a hard plastic mudflap similar to Tom's idea. It looks great no need to paint
and it is tuffer'n elephant butter.
I recently installed new lights
on a friend's utility trailer. I just cut the supplied bracket off (easily done with my razor knife) and made a new stronger one.
The one's which come with trailer lights
are not impressive, in fact most trailer lights
supplied on RV trailers are kinda pathetic to say the least.