HAHAHA! I just did this shit 4 times my last track day in my c6z...And Ill probably be doing it again this Friday when I go back. Only way around it is a longer trailer and you back the car on to have fender clearance OR raise the whole deck up with 2x12s or whatever
Or buy a trailer that has the fold down mud/splash guards such as what U-haul uses. Then you don't need to worry about opening or closing doors with them in the way.