After having worked in the kitchen for 2 months, I would now go ahead and send it back. Normally I would say to just eat it and that it is your mistake, but after working in the kitchen, I know now it is no big deal when something gets sent back.
I have had things sent back and seen things sent back for all kinds of crazy reasons, in just 2 months, so I imagine after a year or so of cooking for a living, the cook just stops caring what your complaint is and why you are sending it back, and just wants to fix it and send it back out to you again fast before it interrupts the other orders.
I had some cheese garlic bread sent back, because the customer wanted it with the cheese melted, the thing was, the cheese was already melted, it even had bubble holes in it from when it had boiled as it melted under the grill. I just changed it to another plate, gave it 30 seconds in the microwave and sent it back out, and was told “it was very nice thank you” by the waitress on behalf of the clients.
With it being a burger, that you want to be more cooked, there is no problem in sending it back. If you sent back that order to me, I would just flip open the burger, put the meat back on the grill for a minute each side or so, then sent it back, problem solved.