I've done smoked burgers on a couple of occasions and while I loved the flavor I always thought the texture left something to be desired unless you followed up with a high heat finishing step. I did them at 215 and I pulled them at 135-140 and then reverse seared over medium-high heat in a preheated cast iron skillet with a little bacon fat (butter would also work). Hickory dowels were my wood of choice for this though I've also used pecan with good results.
I think the grill would do what you need as well but I've always preferred to keep the burgers on a flattop or a skillet to let the juices sear into a nice crust. No matter what, the secondary cooking step will get you up to 145-150 (your upper limit for medium) quickly so keep a close eye on your burgers.
The smoke won't be overwhelming and allow them to stay quite juicy. I did let some worcestershire sauce and spices soak into the meat before putting into the smoker so that may have helped a bit. I've also rubbed them with Pickapeppa Sauce or A-1 Thick & Hearty before smoking as well and both flavors were great.