It's impossible to tell you how much it will cost without more information (car type, which door, pictures of damage, etc).
I'd go get a quote from a good body shop before you do anything.  They'll be able to tell you if the door really needs to be replaced and if there is any other damage that will need to be taken care of.  Honestly, I'd pay out of pocket if its just a few hundred past the deductible but mostly because its less headache.  Any time you deal with insurance it becomes more complicated (have to use certain shops, usually have less control over exactly how they do the work on your car, less leverage on getting the car finished quickly, etc) .