Would it be advisable to partially pre-grill hamburgers and freeze so that they can be pan fried when a customer orders?
I understand the challenges of running a small restaurant, I've experienced the extremes of the problems. During tourist season I always hired help, but during winter it was just me - Bartender, waitress, cook, dishwasher, bookkeeper, janitor - you name it. Occasionally buses would pull up and unload 30 people on me at once, of course I was never prepped for that. I had some tricks up my sleeve just for that occurrence. 

As far as par-cooking your burgers in advance, don't do it. You will lose juiciness, I know this from experience. Besides, if you fully cook par-cooked burgers from frozen, you're really not saving any time after they're ordered. Grill marks don't matter, you don't see them anyway. You're much better off being sure that the patties are of excellent quality to begin with.

The scope of your question is such that all I can really say is, "Don't do it." I do have about about a gazillion things I could share with you that might help speed up your service, minimize waste and put out a great product. Meet me in chat, I'm here quite a bit. You could also write questions that are a bit more broad. I'd love to help in a meaningful way.